Inverse trigonometric functions

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vector-toc-level-3"> <a class="vector-toc-link" href="#Detailed_example_and_explanation_of_the_"plus_or_minus"_symbol_±"> <div class="vector-toc-text"> 2.2.1 Detailed example and explanation of the "plus or minus" symbol ± </div> </a> <ul id="toc-Detailed_example_and_explanation_of_the_"plus_or_minus"_symbol_±-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equal_identical_trigonometric_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Equal_identical_trigonometric_functions"> <div class="vector-toc-text"> 2.2.2 Equal identical trigonometric functions </div> </a> <ul id="toc-Equal_identical_trigonometric_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transforming_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transforming_equations"> <div class="vector-toc-text"> 2.3 Transforming equations </div> </a> <ul id="toc-Transforming_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships_between_trigonometric_functions_and_inverse_trigonometric_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationships_between_trigonometric_functions_and_inverse_trigonometric_functions"> <div class="vector-toc-text"> 2.4 Relationships between trigonometric functions and inverse trigonometric functions </div> </a> <ul id="toc-Relationships_between_trigonometric_functions_and_inverse_trigonometric_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships_among_the_inverse_trigonometric_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationships_among_the_inverse_trigonometric_functions"> <div class="vector-toc-text"> 2.5 Relationships among the inverse trigonometric functions </div> </a> <ul id="toc-Relationships_among_the_inverse_trigonometric_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arctangent_addition_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arctangent_addition_formula"> <div class="vector-toc-text"> 2.6 Arctangent addition formula </div> </a> <ul id="toc-Arctangent_addition_formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_calculus"> <div class="vector-toc-text"> 3 In calculus </div> </a> <button aria-controls="toc-In_calculus-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle In calculus subsection </button> <ul id="toc-In_calculus-sublist" class="vector-toc-list"> <li id="toc-Derivatives_of_inverse_trigonometric_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivatives_of_inverse_trigonometric_functions"> <div class="vector-toc-text"> 3.1 Derivatives of inverse trigonometric functions </div> </a> <ul id="toc-Derivatives_of_inverse_trigonometric_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expression_as_definite_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expression_as_definite_integrals"> <div class="vector-toc-text"> 3.2 Expression as definite integrals </div> </a> <ul id="toc-Expression_as_definite_integrals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_series"> <div class="vector-toc-text"> 3.3 Infinite series </div> </a> <ul id="toc-Infinite_series-sublist" class="vector-toc-list"> <li id="toc-Continued_fractions_for_arctangent" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Continued_fractions_for_arctangent"> <div class="vector-toc-text"> 3.3.1 Continued fractions for arctangent </div> </a> <ul id="toc-Continued_fractions_for_arctangent-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Indefinite_integrals_of_inverse_trigonometric_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Indefinite_integrals_of_inverse_trigonometric_functions"> <div class="vector-toc-text"> 3.4 Indefinite integrals of inverse trigonometric functions </div> </a> <ul id="toc-Indefinite_integrals_of_inverse_trigonometric_functions-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 3.4.1 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Extension_to_the_complex_plane" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extension_to_the_complex_plane"> <div class="vector-toc-text"> 4 Extension to the complex plane </div> </a> <button aria-controls="toc-Extension_to_the_complex_plane-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Extension to the complex plane subsection </button> <ul id="toc-Extension_to_the_complex_plane-sublist" class="vector-toc-list"> <li id="toc-Logarithmic_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_forms"> <div class="vector-toc-text"> 4.1 Logarithmic forms </div> </a> <ul id="toc-Logarithmic_forms-sublist" class="vector-toc-list"> <li id="toc-Generalization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Generalization"> <div class="vector-toc-text"> 4.1.1 Generalization </div> </a> <ul id="toc-Generalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_proof" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_proof"> <div class="vector-toc-text"> 4.1.2 Example proof </div> </a> <ul id="toc-Example_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Finding_the_angle_of_a_right_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finding_the_angle_of_a_right_triangle"> <div class="vector-toc-text"> 5.1 Finding the angle of a right triangle </div> </a> <ul id="toc-Finding_the_angle_of_a_right_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_computer_science_and_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_computer_science_and_engineering"> <div class="vector-toc-text"> 5.2 In computer science and engineering </div> </a> <ul id="toc-In_computer_science_and_engineering-sublist" class="vector-toc-list"> <li id="toc-Two-argument_variant_of_arctangent" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Two-argument_variant_of_arctangent"> <div class="vector-toc-text"> 5.2.1 Two-argument variant of arctangent </div> </a> <ul id="toc-Two-argument_variant_of_arctangent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arctangent_function_with_location_parameter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Arctangent_function_with_location_parameter"> <div class="vector-toc-text"> 5.2.2 Arctangent function with location parameter </div> </a> <ul id="toc-Arctangent_function_with_location_parameter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_accuracy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Numerical_accuracy"> <div class="vector-toc-text"> 5.2.3 Numerical accuracy </div> </a> <ul id="toc-Numerical_accuracy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 6 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 7 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 9 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Inverse trigonometric functions</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 53 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-53" aria-hidden="true" > 53 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D8%A7%D9%84_%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9_%D8%B9%D9%83%D8%B3%D9%8A%D8%A9" title="دوال مثلثية عكسية – Arabic" lang="ar" hreflang="ar" data-title="دوال مثلثية عكسية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/T%C9%99rs_triqonometrik_funksiyalar" title="Tərs triqonometrik funksiyalar – Azerbaijani" lang="az" hreflang="az" data-title="Tərs triqonometrik funksiyalar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%BF%E0%A6%AA%E0%A6%B0%E0%A7%80%E0%A6%A4_%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8B%E0%A6%A3%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="বিপরীত ত্রিকোণমিতিক অপেক্ষক – Bangla" lang="bn" hreflang="bn" data-title="বিপরীত ত্রিকোণমিতিক অপেক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ge%CC%8Dk-sa%E2%81%BF-kak_h%C3%A2m-s%C3%B2%CD%98" title="Ge̍k-saⁿ-kak hâm-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Ge̍k-saⁿ-kak hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B8%D1%80%D0%B5_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Кире тригонометрик функциялар – Bashkir" lang="ba" hreflang="ba" data-title="Кире тригонометрик функциялар" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B8_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Обратни тригонометрични функции – Bulgarian" lang="bg" hreflang="bg" data-title="Обратни тригонометрични функции" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Inverses_de_les_funcions_trigonom%C3%A8triques" title="Inverses de les funcions trigonomètriques – Catalan" lang="ca" hreflang="ca" data-title="Inverses de les funcions trigonomètriques" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BA%D1%83%D1%82%C4%83%D0%BD%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%81%D0%B5%D0%BC" title="Тригонометрилле кутăнла функцисем – Chuvash" lang="cv" hreflang="cv" data-title="Тригонометрилле кутăнла функцисем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cyklometrick%C3%A1_funkce" title="Cyklometrická funkce – Czech" lang="cs" hreflang="cs" data-title="Cyklometrická funkce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arcus-funktioner" title="Arcus-funktioner – Danish" lang="da" hreflang="da" data-title="Arcus-funktioner" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Arkusfunktion" title="Arkusfunktion – German" lang="de" hreflang="de" data-title="Arkusfunktion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Arkusfunktsioonid" title="Arkusfunktsioonid – Estonian" lang="et" hreflang="et" data-title="Arkusfunktsioonid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CF%84%CE%AF%CF%83%CF%84%CF%81%CE%BF%CF%86%CE%B5%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CE%AD%CF%82_%CF%83%CF%85%CE%BD%CE%B1%CF%81%CF%84%CE%AE%CF%83%CE%B5%CE%B9%CF%82" title="Αντίστροφες τριγωνομετρικές συναρτήσεις – Greek" lang="el" hreflang="el" data-title="Αντίστροφες τριγωνομετρικές συναρτήσεις" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_trigonom%C3%A9trica_inversa" title="Función trigonométrica inversa – Spanish" lang="es" hreflang="es" data-title="Función trigonométrica inversa" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Inversa_trigonometria_funkcio" title="Inversa trigonometria funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Inversa trigonometria funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Alderantzizko_funtzio_trigonometriko" title="Alderantzizko funtzio trigonometriko – Basque" lang="eu" hreflang="eu" data-title="Alderantzizko funtzio trigonometriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D8%A8%D8%B9_%D9%85%D8%B9%DA%A9%D9%88%D8%B3_%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA%DB%8C" title="توابع معکوس مثلثاتی – Persian" lang="fa" hreflang="fa" data-title="توابع معکوس مثلثاتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_circulaire_r%C3%A9ciproque" title="Fonction circulaire réciproque – French" lang="fr" hreflang="fr" data-title="Fonction circulaire réciproque" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3ns_trigonom%C3%A9tricas_inversas" title="Funcións trigonométricas inversas – Galician" lang="gl" hreflang="gl" data-title="Funcións trigonométricas inversas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%AD%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98" title="역삼각 함수 – Korean" lang="ko" hreflang="ko" data-title="역삼각 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%AF%D5%A1%D5%A4%D5%A1%D6%80%D5%B1_%D5%A5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B6%D5%A5%D6%80" title="Հակադարձ եռանկյունաչափական ֆունկցիաներ – Armenian" lang="hy" hreflang="hy" data-title="Հակադարձ եռանկյունաչափական ֆունկցիաներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%B2%E0%A5%8B%E0%A4%AE_%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="प्रतिलोम त्रिकोणमितीय फलन – Hindi" lang="hi" hreflang="hi" data-title="प्रतिलोम त्रिकोणमितीय फलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Arkus_funkcije" title="Arkus funkcije – Croatian" lang="hr" hreflang="hr" data-title="Arkus funkcije" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_invers_trigonometri" title="Fungsi invers trigonometri – Indonesian" lang="id" hreflang="id" data-title="Fungsi invers trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_trigonometrica_inversa" title="Funzione trigonometrica inversa – Italian" lang="it" hreflang="it" data-title="Funzione trigonometrica inversa" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA_%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA_%D7%94%D7%A4%D7%95%D7%9B%D7%95%D7%AA" title="פונקציות טריגונומטריות הפוכות – Hebrew" lang="he" hreflang="he" data-title="פונקציות טריגונומטריות הפוכות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D1%80%D1%96_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D0%B0%D1%80" title="Кері тригонометриялық функциялар – Kazakh" lang="kk" hreflang="kk" data-title="Кері тригонометриялық функциялар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%BA%D1%84%D0%B5%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аркфенкция – Kyrgyz" lang="ky" hreflang="ky" data-title="Аркфенкция" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functiones_trigonometricae_inversae" title="Functiones trigonometricae inversae – Latin" lang="la" hreflang="la" data-title="Functiones trigonometricae inversae" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Invers%C4%81s_trigonometrisk%C4%81s_funkcijas" title="Inversās trigonometriskās funkcijas – Latvian" lang="lv" hreflang="lv" data-title="Inversās trigonometriskās funkcijas" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B8_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Инверзни тригонометриски функции – Macedonian" lang="mk" hreflang="mk" data-title="Инверзни тригонометриски функции" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Cyclometrische_functie" title="Cyclometrische functie – Dutch" lang="nl" hreflang="nl" data-title="Cyclometrische functie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%86%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0" title="逆三角関数 – Japanese" lang="ja" hreflang="ja" data-title="逆三角関数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Inverse_trigonometriske_funksjoner" title="Inverse trigonometriske funksjoner – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Inverse trigonometriske funksjoner" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Arcus-funksjon" title="Arcus-funksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Arcus-funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%93%E1%9E%BB%E1%9E%82%E1%9E%98%E1%9E%93%E1%9F%8D%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E%E1%9E%98%E1%9E%B6%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%85%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%9F%E1%9F%8B" title="អនុគមន៍ត្រីកោណមាត្រច្រាស់ – Khmer" lang="km" hreflang="km" data-title="អនុគមន៍ត្រីកោណមាត្រច្រាស់" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcje_cyklometryczne" title="Funkcje cyklometryczne – Polish" lang="pl" hreflang="pl" data-title="Funkcje cyklometryczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%B5es_trigonom%C3%A9tricas_inversas" title="Funções trigonométricas inversas – Portuguese" lang="pt" hreflang="pt" data-title="Funções trigonométricas inversas" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D1%8B%D0%B5_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Обратные тригонометрические функции – Russian" lang="ru" hreflang="ru" data-title="Обратные тригонометрические функции" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%AD%E0%B7%92%E0%B6%BD%E0%B7%9D%E0%B6%B8_%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%9A%E0%B7%9D%E0%B6%AB%E0%B6%B8%E0%B7%92%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD" title="ප්‍රතිලෝම ත්‍රිකෝණමිතික ශ්‍රිත – Sinhala" lang="si" hreflang="si" data-title="ප්‍රතිලෝම ත්‍රිකෝණමිතික ශ්‍රිත" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cyklometrick%C3%A1_funkcia" title="Cyklometrická funkcia – Slovak" lang="sk" hreflang="sk" data-title="Cyklometrická funkcia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kro%C5%BEna_funkcija" title="Krožna funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Krožna funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95_%DA%BE%DB%95%DA%B5%DA%AF%DB%95%DA%95%D8%A7%D9%88%DB%95%DA%A9%D8%A7%D9%86" title="فانکشنە سێگۆشەیییە ھەڵگەڕاوەکان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="فانکشنە سێگۆشەیییە ھەڵگەڕاوەکان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B2%D0%B5%D1%80%D0%B7%D0%BD%D0%B5_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B5" title="Инверзне тригонометријске функције – Serbian" lang="sr" hreflang="sr" data-title="Инверзне тригонометријске функције" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Inverzne_trigonometrijske_funkcije" title="Inverzne trigonometrijske funkcije – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Inverzne trigonometrijske funkcije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Arkusfunktiot" title="Arkusfunktiot – Finnish" lang="fi" hreflang="fi" data-title="Arkusfunktiot" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81_%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="நேர்மாறு முக்கோணவியல் சார்புகள் – Tamil" lang="ta" hreflang="ta" data-title="நேர்மாறு முக்கோணவியல் சார்புகள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%B5%E0%B9%82%E0%B8%81%E0%B8%93%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%B4%E0%B8%9C%E0%B8%81%E0%B8%9C%E0%B8%B1%E0%B8%99" title="ฟังก์ชันตรีโกณมิติผกผัน – Thai" lang="th" hreflang="th" data-title="ฟังก์ชันตรีโกณมิติผกผัน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ters_trigonometrik_fonksiyonlar" title="Ters trigonometrik fonksiyonlar – Turkish" lang="tr" hreflang="tr" data-title="Ters trigonometrik fonksiyonlar" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D1%80%D0%BD%D0%B5%D0%BD%D1%96_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97" title="Обернені тригонометричні функції – Ukrainian" lang="uk" hreflang="uk" data-title="Обернені тригонометричні функції" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%A1c_h%C3%A0m_l%C6%B0%E1%BB%A3ng_gi%C3%A1c_ng%C6%B0%E1%BB%A3c" title="Các hàm lượng giác ngược – Vietnamese" lang="vi" hreflang="vi" data-title="Các hàm lượng giác ngược" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8F%8D%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8" title="反三角函數 – Cantonese" lang="yue" hreflang="yue" data-title="反三角函數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8D%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0" title="反三角函数 – Chinese" lang="zh" hreflang="zh" data-title="反三角函数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q674533#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" 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function</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Inverse functions of sin, cos, tan, etc.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="background:#ccccff;"><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Sinus_und_Kosinus_am_Einheitskreis_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/250px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/375px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Sinus_und_Kosinus_am_Einheitskreis_1.svg/500px-Sinus_und_Kosinus_am_Einheitskreis_1.svg.png 2x" data-file-width="410" data-file-height="410" /></a></td></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Outline_of_trigonometry" title="Outline of trigonometry">Outline</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">History</a></li> <li><a href="/wiki/Uses_of_trigonometry" title="Uses of trigonometry">Usage</a></li></ul> <ul><li><a href="/wiki/Trigonometric_functions" title="Trigonometric functions">Functions</a> (<a href="/wiki/Sine_and_cosine" title="Sine and cosine">sin</a>, <a href="/wiki/Sine_and_cosine" title="Sine and cosine">cos</a>, <a href="/wiki/Trigonometric_functions#tangent" title="Trigonometric functions">tan</a>, <a class="mw-selflink selflink">inverse</a>)</li> <li><a href="/wiki/Generalized_trigonometry" title="Generalized trigonometry">Generalized trigonometry</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Reference</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">Identities</a></li> <li><a href="/wiki/Exact_trigonometric_values" title="Exact trigonometric values">Exact constants</a></li> <li><a href="/wiki/Trigonometric_tables" title="Trigonometric tables">Tables</a></li> <li><a href="/wiki/Unit_circle" title="Unit circle">Unit circle</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Laws and theorems</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Law_of_sines" title="Law of sines">Sines</a></li> <li><a href="/wiki/Law_of_cosines" title="Law of cosines">Cosines</a></li> <li><a href="/wiki/Law_of_tangents" title="Law of tangents">Tangents</a></li> <li><a href="/wiki/Law_of_cotangents" title="Law of cotangents">Cotangents</a></li></ul> <ul><li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> <a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">Trigonometric substitution</a></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">Integrals</a> (<a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse functions</a>)</li> <li><a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Derivatives</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;"> Mathematicians</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.7em;"> <ul><li><a href="/wiki/Hipparchus" title="Hipparchus">Hipparchus</a></li> <li><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Habash_al-Hasib_al-Marwazi" class="mw-redirect" title="Habash al-Hasib al-Marwazi">al-Hasib</a></li> <li><a href="/wiki/Al-Battani" title="Al-Battani">al-Battani</a></li> <li><a href="/wiki/Regiomontanus" title="Regiomontanus">Regiomontanus</a></li> <li><a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">Viète</a></li> <li><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Fourier</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Trigonometry" title="Template:Trigonometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Trigonometry" title="Template talk:Trigonometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Trigonometry" title="Special:EditPage/Template:Trigonometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Arctangent" redirects here. For the Earthtone9 album, see <a href="/wiki/Arc%27tan%27gent" title="Arc'tan'gent">arc'tan'gent</a>. For the music event, see <a href="/wiki/ArcTanGent_Festival" title="ArcTanGent Festival">ArcTanGent Festival</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the inverse trigonometric functions (occasionally also called antitrigonometric,<a href="#cite_note-Hall_1909-1">[1]</a> cyclometric,<a href="#cite_note-cyclometric-2">[2]</a> or arcus functions<a href="#cite_note-arcus-3">[3]</a>) are the <a href="/wiki/Inverse_function" title="Inverse function">inverse functions</a> of the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, under suitably restricted <a href="/wiki/Domain_of_a_function" title="Domain of a function">domains</a>. Specifically, they are the inverses of the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a>, <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, <a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">tangent</a>, <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a>, <a href="/wiki/Secant_(trigonometry)" class="mw-redirect" title="Secant (trigonometry)">secant</a>, and <a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">cosecant</a> functions,<a href="#cite_note-4">[4]</a> and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Navigation" title="Navigation">navigation</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Geometry" title="Geometry">geometry</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=1" title="Edit section: Notation">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Arcsin_and_arccos_as_actual_arc_lengths.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Arcsin_and_arccos_as_actual_arc_lengths.svg/220px-Arcsin_and_arccos_as_actual_arc_lengths.svg.png" decoding="async" width="220" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Arcsin_and_arccos_as_actual_arc_lengths.svg/330px-Arcsin_and_arccos_as_actual_arc_lengths.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Arcsin_and_arccos_as_actual_arc_lengths.svg/440px-Arcsin_and_arccos_as_actual_arc_lengths.svg.png 2x" data-file-width="191" data-file-height="188" /></a><figcaption>For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Trigonometric_functions#Notation" title="Trigonometric functions">Trigonometric functions § Notation</a></div> Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc.<a href="#cite_note-Hall_1909-1">[1]</a> (This convention is used throughout this article.) This notation arises from the following geometric relationships:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] when measuring in radians, an angle of θ radians will correspond to an <a href="/wiki/Circular_arc" title="Circular arc">arc</a> whose length is rθ, where r is the radius of the circle. Thus in the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians.<a href="#cite_note-Americana_1912-5">[5]</a> In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan.<a href="#cite_note-6">[6]</a> The notations sin−1(x), cos−1(x), tan−1(x), etc., as introduced by <a href="/wiki/John_Herschel" title="John Herschel">John Herschel</a> in 1813,<a href="#cite_note-Cajori-7">[7]</a><a href="#cite_note-Herschel_1813-8">[8]</a> are often used as well in English-language sources,<a href="#cite_note-Hall_1909-1">[1]</a> much more than the also <a href="/wiki/Iterated_function#Definition" title="Iterated function">established</a> sin[−1](x), cos[−1](x), tan[−1](x) – conventions consistent with the notation of an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>, that is useful (for example) to define the <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued</a> version of each inverse trigonometric function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>k</mi> <mo>∣</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f58e4b6ab1d8b36aaa6e0a76d22b3e6200dac0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.01ex; height:3.176ex;" alt="{\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.}"> However, this might appear to conflict logically with the common semantics for expressions such as sin2(x) (although only sin2 x, without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the <a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocal</a> (<a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>) and <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>.<a href="#cite_note-9">[9]</a> The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos(x))−1 = sec(x). Nevertheless, certain authors advise against using it, since it is ambiguous.<a href="#cite_note-Hall_1909-1">[1]</a><a href="#cite_note-Korn_2000-10">[10]</a> Another precarious convention used by a small number of authors is to use an <a href="/wiki/UPPERCASE" class="mw-redirect" title="UPPERCASE">uppercase</a> first letter, along with a “−1” superscript: Sin−1(x), Cos−1(x), Tan−1(x), etc.<a href="#cite_note-Bhatti_1999-11">[11]</a> Although it is intended to avoid confusion with the <a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocal</a>, which should be represented by sin−1(x), cos−1(x), etc., or, better, by sin−1 x, cos−1 x, etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> and <a href="/wiki/Magma_(computer_algebra_system)" title="Magma (computer algebra system)">MAGMA</a>) use those very same capitalised representations for the standard trig functions, whereas others (<a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="/wiki/SymPy" title="SymPy">SymPy</a>, <a href="/wiki/NumPy" title="NumPy">NumPy</a>, <a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a>, <a href="/wiki/Maple_(software)" title="Maple (software)">MAPLE</a>, etc.) use lower-case. Hence, since 2009, the <a href="/wiki/ISO_80000-2#Part_2:_Mathematics" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> standard has specified solely the "arc" prefix for the inverse functions. <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=2" title="Edit section: Basic concepts">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TrigFunctionDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/220px-TrigFunctionDiagram.svg.png" decoding="async" width="220" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/330px-TrigFunctionDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/TrigFunctionDiagram.svg/440px-TrigFunctionDiagram.svg.png 2x" data-file-width="644" data-file-height="655" /></a><figcaption>The points labelled <style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Sec(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Csc(θ) represent the length of the line segment from the origin to that point. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Sin(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Tan(θ), and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">1 are the heights to the line starting from the x-axis, while <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cos(θ), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">1, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">Cot(θ) are lengths along the x-axis starting from the origin.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Principal_values">Principal values</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=3" title="Edit section: Principal values">edit</a>]</div> Since none of the six trigonometric functions are <a href="/wiki/One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one</a>, they must be restricted in order to have inverse functions. Therefore, the result <a href="/wiki/Range_of_a_function" title="Range of a function">ranges</a> of the inverse functions are proper (i.e. strict) <a href="/wiki/Subset" title="Subset">subsets</a> of the domains of the original functions. For example, using function in the sense of <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued functions</a>, just as the <a href="/wiki/Square_root" title="Square root">square root</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\sqrt {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a6e259ae203e3c564ba35d228d13365f846a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.52ex; height:3.009ex;" alt="{\displaystyle y={\sqrt {x}}}"> could be defined from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c761e4a192386ab0f1c0875666e02691ee0790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.29ex; height:3.009ex;" alt="{\displaystyle y^{2}=x,}"> the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e0923cd97e3e42f303d625dde6e5b468836824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.355ex; height:2.843ex;" alt="{\displaystyle y=\arcsin(x)}"> is defined so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(y)=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(y)=x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3859a594fdce25e40aadc13377cc3d67c27128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.895ex; height:2.843ex;" alt="{\displaystyle \sin(y)=x.}"> For a given real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ac715148ec5700e92bbb11a23145e493c8df0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.306ex; height:2.509ex;" alt="{\displaystyle -1\leq x\leq 1,}"> there are multiple (in fact, <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinitely</a> many) numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(y)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(y)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5161e8df56915fc965895edb209b1800d8c69b32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.249ex; height:2.843ex;" alt="{\displaystyle \sin(y)=x}">; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(0)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(0)=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb58f6a6ab2cdb83581b1e71db53c00978fb872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.735ex; height:2.843ex;" alt="{\displaystyle \sin(0)=0,}"> but also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi )=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi )=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49110e063e5c6c1b7a54e2680a4f24541009ca58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.905ex; height:2.843ex;" alt="{\displaystyle \sin(\pi )=0,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2\pi )=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\pi )=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0acabc47989df7c446209b84e7edbd7f18886b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.067ex; height:2.843ex;" alt="{\displaystyle \sin(2\pi )=0,}"> etc. When only one value is desired, the function may be restricted to its <a href="/wiki/Principal_branch" title="Principal branch">principal branch</a>. With this restriction, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in the domain, the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f111f8c1e5de2f92ee17eeabd64c4ea1bcd55196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle \arcsin(x)}"> will evaluate only to a single value, called its <a href="/wiki/Principal_value" title="Principal value">principal value</a>. These properties apply to all the inverse trigonometric functions. The principal inverses are listed in the following table. <table class="wikitable" style="text-align:center"> <tbody><tr> <th>Name </th> <th>Usual notation </th> <th>Definition </th> <th>Domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> for real result </th> <th>Range of usual principal value (<a href="/wiki/Radian" title="Radian">radians</a>) </th> <th>Range of usual principal value (<a href="/wiki/Degree_(angle)" title="Degree (angle)">degrees</a>) </th></tr> <tr> <td>arcsine</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e0923cd97e3e42f303d625dde6e5b468836824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.355ex; height:2.843ex;" alt="{\displaystyle y=\arcsin(x)}"></td> <td>x = <a href="/wiki/Sine" class="mw-redirect" title="Sine">sin</a>(y)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efafb1db8d9f68062b352591b873384c668863a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.497ex; height:4.676ex;" alt="{\displaystyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -90^{\circ }\leq y\leq 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -90^{\circ }\leq y\leq 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c6ea7f515defdcb5c8da0113a0043d7cf58153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.919ex; height:2.676ex;" alt="{\displaystyle -90^{\circ }\leq y\leq 90^{\circ }}"> </td></tr> <tr> <td>arccosine</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180deffe95226787d18a945b5093e1a9f64d85b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.611ex; height:2.843ex;" alt="{\displaystyle y=\arccos(x)}"></td> <td>x = <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cos</a>(y)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq y\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq y\leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefa00e53d95829eab873c055bad5904bbdab2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.847ex; height:2.509ex;" alt="{\displaystyle 0\leq y\leq \pi }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }\leq y\leq 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }\leq y\leq 180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3f0a77577d34c5b82f33a39f0ff0622e27fdc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.111ex; height:2.676ex;" alt="{\displaystyle 0^{\circ }\leq y\leq 180^{\circ }}"> </td></tr> <tr> <td>arctangent</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arctan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arctan(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ab78600eb69cbc1af4ae58af2532a134b99275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.859ex; height:2.843ex;" alt="{\displaystyle y=\arctan(x)}"></td> <td>x = <a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">tan</a>(y)</td> <td>all real numbers</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abf498afbd5441808be418f8e62ef4969fa055b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.497ex; height:4.676ex;" alt="{\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -90^{\circ }<y<90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>y</mi> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -90^{\circ }<y<90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339dbbcb0a94d0d66862d23af19f2ea6fc8999bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.919ex; height:2.676ex;" alt="{\displaystyle -90^{\circ }<y<90^{\circ }}"> </td></tr> <tr> <td>arccotangent</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arccot}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arccot}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f2eae39c3824b4fd81b90e7cfb611348459cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.599ex; height:2.843ex;" alt="{\displaystyle y=\operatorname {arccot}(x)}"></td> <td>x = <a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cot</a>(y)</td> <td>all real numbers </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<y<\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<y<\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c651b3e30c00578bf38fc059d5d636235048d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.847ex; height:2.509ex;" alt="{\displaystyle 0<y<\pi }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }<y<180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>y</mi> <mo><</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }<y<180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b260cfda7080b7f3c33bfe6519c7876951e3491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.111ex; height:2.676ex;" alt="{\displaystyle 0^{\circ }<y<180^{\circ }}"> </td></tr> <tr> <td>arcsecant</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arcsec}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arcsec}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6d1afcbf210c594d8753989476595c6fba955b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.48ex; height:2.843ex;" alt="{\displaystyle y=\operatorname {arcsec}(x)}"></td> <td>x = <a href="/wiki/Secant_(trigonometry)" class="mw-redirect" title="Secant (trigonometry)">sec</a>(y)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left\vert x\right\vert }\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left\vert x\right\vert }\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77b44fbf3df91e0a9e60819d62e88d33db8f8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle {\left\vert x\right\vert }\geq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320a53e015f032429effa8de8ea993d8ffe49435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.771ex; height:4.676ex;" alt="{\displaystyle 0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }\leq y<90^{\circ }{\text{ or }}90^{\circ }<y\leq 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>≤</mo> <mi>y</mi> <mo><</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>y</mi> <mo>≤</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }\leq y<90^{\circ }{\text{ or }}90^{\circ }<y\leq 180^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9610764127f1934fc646c3d3e49344a7fa7b02f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.457ex; height:2.676ex;" alt="{\displaystyle 0^{\circ }\leq y<90^{\circ }{\text{ or }}90^{\circ }<y\leq 180^{\circ }}"> </td></tr> <tr> <td>arccosecant</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\operatorname {arccsc}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\operatorname {arccsc}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30409f5bec403c8e5ab808e8fe9a23896f234673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.48ex; height:2.843ex;" alt="{\displaystyle y=\operatorname {arccsc}(x)}"></td> <td>x = <a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">csc</a>(y)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left\vert x\right\vert }\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left\vert x\right\vert }\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77b44fbf3df91e0a9e60819d62e88d33db8f8c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle {\left\vert x\right\vert }\geq 1}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{2}}\leq y<0{\text{ or }}0<y\leq {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{2}}\leq y<0{\text{ or }}0<y\leq {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e3596f00cdf9c700f9adc362289565962aba89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.41ex; height:4.676ex;" alt="{\displaystyle -{\frac {\pi }{2}}\leq y<0{\text{ or }}0<y\leq {\frac {\pi }{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -90^{\circ }\leq y<0^{\circ }{\text{ or }}0^{\circ }<y\leq 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>≤</mo> <mi>y</mi> <mo><</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo><</mo> <mi>y</mi> <mo>≤</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -90^{\circ }\leq y<0^{\circ }{\text{ or }}0^{\circ }<y\leq 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ffd3f991cadb3560c97c37036c2ec8e7c88337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.94ex; height:2.676ex;" alt="{\displaystyle -90^{\circ }\leq y<0^{\circ }{\text{ or }}0^{\circ }<y\leq 90^{\circ }}"> </td></tr> </tbody></table> Note: Some authors [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] define the range of arcsecant to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\pi \leq y<{\frac {3\pi }{2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>π</mi> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\pi \leq y<{\frac {3\pi }{2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26465e3e1c6e6c01a539cebe1caa82450ff3c3d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.622ex; height:3.509ex;" alt="{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\pi \leq y<{\frac {3\pi }{2}})}">, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3d107ff09672c3f11393d9fda8f93b75724148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.851ex; height:3.509ex;" alt="{\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},}"> whereas with the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce2ef31322670a3ca77a4ef89427c1f39d41565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.8ex; height:3.176ex;" alt="{\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi )}">, we would have to write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e71e595045794255b2241d8088edad989c8c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.659ex; height:3.509ex;" alt="{\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},}"> since tangent is nonnegative on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0\leq y<{\frac {\pi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>y</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0\leq y<{\frac {\pi }{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d106a05cb24bc3bbddbd0954129ceb26e62001a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.94ex; height:3.176ex;" alt="{\textstyle 0\leq y<{\frac {\pi }{2}},}"> but nonpositive on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {\pi }{2}}<y\leq \pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\pi }{2}}<y\leq \pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da4eb08aeeea6249a58c63d6b94464bb1f92021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.109ex; height:3.176ex;" alt="{\textstyle {\frac {\pi }{2}}<y\leq \pi .}"> For a similar reason, the same authors define the range of arccosecant to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (-\pi <y\leq -{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (-\pi <y\leq -{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11c853471ea0d0e469d62ff9117cbfbe4ca86b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.983ex; height:3.176ex;" alt="{\textstyle (-\pi <y\leq -{\frac {\pi }{2}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 0<y\leq {\frac {\pi }{2}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 0<y\leq {\frac {\pi }{2}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a48d113e0e4f24b2b4ccbd93cf728fecd03fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.844ex; height:3.176ex;" alt="{\textstyle 0<y\leq {\frac {\pi }{2}}).}"> <div class="mw-heading mw-heading4"><h4 id="Domains">Domains</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=4" title="Edit section: Domains">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is allowed to be a <a href="/wiki/Complex_number" title="Complex number">complex number</a>, then the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> applies only to its real part. The table below displays names and domains of the inverse trigonometric functions along with the <a href="/wiki/Image_of_a_function" class="mw-redirect" title="Image of a function">range</a> of their usual <a href="/wiki/Principal_value" title="Principal value">principal values</a> in <a href="/wiki/Radians" class="mw-redirect" title="Radians">radians</a>. <table class="wikitable" style="text-align: center; ;"> <tbody><tr> <th style="border-style: solid none solid solid;">Name </th> <th style="border-style: solid none;">Symbol </th> <th style="border-style: solid none;"> </th> <th style="border-style: solid none;"><a href="/wiki/Domain_of_a_function" title="Domain of a function">Domain</a> </th> <th style="border-style: solid none;"> </th> <th style="border-style: solid none;"><a href="/wiki/Image_of_a_function" class="mw-redirect" title="Image of a function">Image/Range</a> </th> <th style="border-style: solid none solid solid;">Inverse function </th> <th style="border-style: solid none;"> </th> <th style="border-style: solid none;">Domain </th> <th style="border-style: solid none;"> </th> <th style="border-style: solid none;"><a href="/wiki/Image_of_a_function" class="mw-redirect" title="Image of a function">Image</a> of <a href="/wiki/Principal_value" title="Principal value">principal values</a> </th></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627fcadc0495786c9167c5487a1a795bb1940edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.962ex; height:2.176ex;" alt="{\displaystyle \arcsin }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2052f9d4a9c6a14f6db2e4bcd2606bce26d720d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.337ex; height:3.343ex;" alt="{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]}"> </td></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f10121c1f582201ae32a56303e36bee4191336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.218ex; height:1.676ex;" alt="{\displaystyle \arccos }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}"> </td></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">tangent</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57c91704d9ab0366a6436869e2968491efc5155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ba332f091ff0f76f8bf7e0f3b8d27f87dc2752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.251ex; height:3.343ex;" alt="{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7b430086dd06a52623bbaa883d86baf9811554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.466ex; height:2.009ex;" alt="{\displaystyle \arctan }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d595e79cc45668c2aa4b170a1c2facaf5fdfff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.528ex; height:3.343ex;" alt="{\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> </td></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Cotangent" class="mw-redirect" title="Cotangent">cotangent</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bce4c1822b49c4e7ecd7b9aa07f2fcb1706817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:2.009ex;" alt="{\displaystyle \cot }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} +(0,\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} +(0,\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14fa8fa8a322aed22315d605e210136a40d3326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.06ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} +(0,\pi )}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1975aeae325af753f7a94c5158ea82fe807ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.206ex; height:2.009ex;" alt="{\displaystyle \operatorname {arccot} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f49c5536940f39cd5253d78d969259b7ed1db3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.338ex; height:2.843ex;" alt="{\displaystyle (0,\pi )}"> </td></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Secant_(trigonometry)" class="mw-redirect" title="Secant (trigonometry)">secant</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd93d9784c55e8ef40f81993114bd3c3d01084c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \sec }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ba332f091ff0f76f8bf7e0f3b8d27f87dc2752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.251ex; height:3.343ex;" alt="{\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus (-1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42b9cee82143057af81b0c6beb8022c595d10ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.849ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus (-1,1)}"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482d42102d686f0ca84aa0ef1aba041079968050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.088ex; height:1.676ex;" alt="{\displaystyle \operatorname {arcsec} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus (-1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42b9cee82143057af81b0c6beb8022c595d10ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.849ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus (-1,1)}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>π</mi> <mspace width="thinmathspace" /> <mo stretchy="false">]</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo class="MJX-variant">∖</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c913c2c78d11f57ccd118976bfb4b0595e5a2e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.86ex; height:3.343ex;" alt="{\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}}"> </td></tr> <tr> <td style="border-style: solid none solid solid; text-align: left; padding-left: 0.5em;"><a href="/wiki/Cosecant" class="mw-redirect" title="Cosecant">cosecant</a> </td> <td style="border-style: solid none; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399542cf3e9a873371132ffebe579187d48962ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \csc }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} +(0,\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} +(0,\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14fa8fa8a322aed22315d605e210136a40d3326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.06ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} +(0,\pi )}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus (-1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42b9cee82143057af81b0c6beb8022c595d10ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.849ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus (-1,1)}"> </td> <td style="border-style: solid none solid solid; text-align: right; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d28252695f7581278866d73dc45eff02c9e55fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.088ex; height:1.676ex;" alt="{\displaystyle \operatorname {arccsc} }"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}"> </td> <td style="border-style: solid none; text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus (-1,1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42b9cee82143057af81b0c6beb8022c595d10ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.849ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus (-1,1)}"> </td> <td style="border-style: solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> </td> <td style="border-style: solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>]</mo> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84fda7925743a03ffa0aec3fbed76a2967a3012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.02ex; height:3.343ex;" alt="{\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}"> </td></tr></tbody></table> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} =(-\infty ,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} =(-\infty ,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8788271da67428c1e34135687f5d80d7d2e2e4b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.075ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} =(-\infty ,\infty )}"> denotes the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f507ba45f6f298f158bf3282d3ab053f90a7e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.762ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}}"> denotes the set of all <a href="/wiki/Integer" title="Integer">integers</a>. The set of all integer multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is denoted by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mtext> </mtext> <mo>:=</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mi>π</mi> <mi>n</mi> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ba02fbfe3d901c80a1c89d4e17758ec57f4821" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.877ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.}"> The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\setminus \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo class="MJX-variant">∖</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\setminus \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c95acc7ee899beef090bb68e3a0e52d116025f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.937ex; height:2.843ex;" alt="{\displaystyle \,\setminus \,}"> denotes <a href="/wiki/Set_subtraction" class="mw-redirect" title="Set subtraction">set subtraction</a> so that, for instance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/105fc2887189c9dbf0d165542a768dcd97f03069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.29ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )}"> is the set of points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> (that is, real numbers) that are not in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7ba2d40fe02b8353711b1882e50b5a868f467c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.623ex; height:2.843ex;" alt="{\displaystyle (-1,1).}"> The <a href="/wiki/Minkowski_sum" class="mw-redirect" title="Minkowski sum">Minkowski sum</a> notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi \mathbb {Z} +(0,\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi \mathbb {Z} +(0,\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ddf389f783e835d9545fdc46d1effb808839dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.06ex; height:2.843ex;" alt="{\textstyle \pi \mathbb {Z} +(0,\pi )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da10badd7d656a0cc45edf97e9111c0e3d39c0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.251ex; height:3.343ex;" alt="{\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}}"> that is used above to concisely write the domains of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>,</mo> <mi>csc</mi> <mo>,</mo> <mi>tan</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>sec</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4655ee6f468cc24b6371fa1ce1c53f0b5fc1683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.819ex; height:2.509ex;" alt="{\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec }"> is now explained. Domain of cotangent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bce4c1822b49c4e7ecd7b9aa07f2fcb1706817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:2.009ex;" alt="{\displaystyle \cot }"> and cosecant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399542cf3e9a873371132ffebe579187d48962ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \csc }">: The domains of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cot \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>cot</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cot \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff7717bbd9cd64ad3032553ac32ea6c80a8c812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.874ex; height:2.009ex;" alt="{\displaystyle \,\cot \,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\csc \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>csc</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\csc \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8d249ee4b378fbf10b4af465eeb3457843e3f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.755ex; height:1.676ex;" alt="{\displaystyle \,\csc \,}"> are the same. They are the set of all angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> at which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta \neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \neq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdaeef26320c87c9125896f13a8a19187559a01a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.241ex; height:2.676ex;" alt="{\displaystyle \sin \theta \neq 0,}"> i.e. all real numbers that are not of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc23a9af36c461617b6369530b647063ac995f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.727ex; height:1.676ex;" alt="{\displaystyle \pi n}"> for some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mo>−</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba156929fc2f50d6e6f447504b9c4e4113a12815" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.709ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}}"> Domain of tangent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57c91704d9ab0366a6436869e2968491efc5155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }"> and secant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd93d9784c55e8ef40f81993114bd3c3d01084c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \sec }">: The domains of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\tan \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>tan</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\tan \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4da3b39d79545cbd9c5909d77838b56eacd0ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.009ex;" alt="{\displaystyle \,\tan \,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\sec \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>sec</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\sec \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8dfde42c859f38a642bd6e904a1e6345e603e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.755ex; height:1.676ex;" alt="{\displaystyle \,\sec \,}"> are the same. They are the set of all angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> at which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta \neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta \neq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e57505d5fc59f61267cf177f38981cf1046e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.676ex;" alt="{\displaystyle \cos \theta \neq 0,}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>⋯</mo> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∪</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a69837358933c08113f878e09b863ac46c3e66c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.806ex; margin-bottom: -0.199ex; width:61.105ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Solutions_to_elementary_trigonometric_equations">Solutions to elementary trigonometric equations</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=5" title="Edit section: Solutions to elementary trigonometric equations">edit</a>]</div> Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3812fa876184aea90aaace4c2a55f8dd740f072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.786ex; height:2.176ex;" alt="{\displaystyle 2\pi :}"> <ul><li>Sine and cosecant begin their period at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k-{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k-{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cfb03ba51e78fd49b03ef0bc9b69b51c41de4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.324ex; height:3.176ex;" alt="{\textstyle 2\pi k-{\frac {\pi }{2}}}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is an integer), finish it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {\pi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {\pi }{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7b79df48d3122060fcb138c79747b31b621bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.971ex; height:3.176ex;" alt="{\textstyle 2\pi k+{\frac {\pi }{2}},}"> and then reverse themselves over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9e3f6e3525f96e4752e5f86aa6b06eafeb5d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.324ex; height:3.176ex;" alt="{\textstyle 2\pi k+{\frac {\pi }{2}}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {3\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {3\pi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4126be00f9518a60465aa570530b8787f3ed9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.793ex; height:3.509ex;" alt="{\textstyle 2\pi k+{\frac {3\pi }{2}}.}"></li> <li>Cosine and secant begin their period at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a44c3f8d9c73506055d859b2bfc314586a9e055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.353ex; height:2.509ex;" alt="{\displaystyle 2\pi k,}"> finish it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004e39b03a48a547a969358f75c3a444a0c2b4bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.525ex; height:2.343ex;" alt="{\displaystyle 2\pi k+\pi .}"> and then reverse themselves over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4706b5f6db2fc51fb3bc606d95ebefd19ddd5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.878ex; height:2.343ex;" alt="{\displaystyle 2\pi k+\pi }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+2\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+2\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1185485729baa845a6cd7bc2e179fe3f76a520de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.687ex; height:2.343ex;" alt="{\displaystyle 2\pi k+2\pi .}"></li> <li>Tangent begins its period at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k-{\frac {\pi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k-{\frac {\pi }{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b971a456fc5fc9d1715da45a1f2d406aadb4c2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.971ex; height:3.176ex;" alt="{\textstyle 2\pi k-{\frac {\pi }{2}},}"> finishes it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {\pi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {\pi }{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7b79df48d3122060fcb138c79747b31b621bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.971ex; height:3.176ex;" alt="{\textstyle 2\pi k+{\frac {\pi }{2}},}"> and then repeats it (forward) over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9e3f6e3525f96e4752e5f86aa6b06eafeb5d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.324ex; height:3.176ex;" alt="{\textstyle 2\pi k+{\frac {\pi }{2}}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi k+{\frac {3\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2\pi k+{\frac {3\pi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4126be00f9518a60465aa570530b8787f3ed9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.793ex; height:3.509ex;" alt="{\textstyle 2\pi k+{\frac {3\pi }{2}}.}"></li> <li>Cotangent begins its period at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a44c3f8d9c73506055d859b2bfc314586a9e055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.353ex; height:2.509ex;" alt="{\displaystyle 2\pi k,}"> finishes it at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+\pi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>π</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+\pi ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3b3ac8474c320137011f9d5aa62ac013b7f83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.525ex; height:2.509ex;" alt="{\displaystyle 2\pi k+\pi ,}"> and then repeats it (forward) over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4706b5f6db2fc51fb3bc606d95ebefd19ddd5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.878ex; height:2.343ex;" alt="{\displaystyle 2\pi k+\pi }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi k+2\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi k+2\pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1185485729baa845a6cd7bc2e179fe3f76a520de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.687ex; height:2.343ex;" alt="{\displaystyle 2\pi k+2\pi .}"></li></ul> This periodicity is reflected in the general inverses, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is some integer. The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a45ee89f965ec5498b347cb54f45243ef88b218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.509ex;" alt="{\displaystyle \theta ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250644a0f511e9078be6f89ba78a606a0e08c0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.695ex; height:2.009ex;" alt="{\displaystyle r,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5748cdb81bf00075de8e7e6828c343687513830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.009ex;" alt="{\displaystyle s,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> all lie within appropriate ranges so that the relevant expressions below are <a href="/wiki/Well-defined" class="mw-redirect" title="Well-defined">well-defined</a>. Note that "for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">" is just another way of saying "for some <a href="/wiki/Integer" title="Integer">integer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb6778a29f576eb23da1dbddffb73b2571359ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k.}">" The symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> is <a href="/wiki/Logical_equality" title="Logical equality">logical equality</a> and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote<a href="#cite_note-12">[note 1]</a> for more details and an example illustrating this concept). <table class="wikitable" style="border: none;"> <caption> </caption> <tbody><tr> <th>Equation</th> <th><a href="/wiki/If_and_only_if" title="If and only if">if and only if</a></th> <th colspan="7">Solution </th></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d80b5d58bed05745127a4e7956b44cb9d9f1e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.587ex; height:2.509ex;" alt="{\displaystyle \sin \theta =y}"> </td> <td style="text-align: center;"><a href="/wiki/Logical_equality" title="Logical equality"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></a> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55e28966ebb251c079bb52f4afd28beff4fb898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.869ex; height:3.176ex;" alt="{\displaystyle (-1)^{k}}"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd36739626506755f4e3b4810134f31af5c8af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.927ex; height:2.843ex;" alt="{\displaystyle \arcsin(y)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786f91f9500bb23dd6f8af17f6bfbbbfe861a880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.606ex; height:2.176ex;" alt="{\displaystyle \csc \theta =r}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55e28966ebb251c079bb52f4afd28beff4fb898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.869ex; height:3.176ex;" alt="{\displaystyle (-1)^{k}}"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16dd3fb8d8203600db048c8d29be520f42e1ed4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.945ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(r)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b477a458116cee4f993321ba6c9da01b28f335f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.195ex; height:2.176ex;" alt="{\displaystyle \pm \,}"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa15507445b7175a6143e9d16ba98a9849374c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.357ex; height:2.843ex;" alt="{\displaystyle \arccos(x)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/025c4b681de7464f3021389a4079c30acf4dba15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.606ex; height:2.176ex;" alt="{\displaystyle \sec \theta =r}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b477a458116cee4f993321ba6c9da01b28f335f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.195ex; height:2.176ex;" alt="{\displaystyle \pm \,}"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8251b85aa038ff6e0f8014f52607d9b0916c58e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.945ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(r)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f078184d98dd7239fc2cb3ca550367c2d43bb9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.026ex; height:2.176ex;" alt="{\displaystyle \tan \theta =s}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651278de756d76d27c88ea01532e061594d33c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.366ex; height:2.843ex;" alt="{\displaystyle \arctan(s)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717f947fc2c5e6e64f8b79fddbc7f110c23c6ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.724ex; height:2.176ex;" alt="{\displaystyle \cot \theta =r}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9b179b91bf601dd3192389bca2afe1533a1e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.931ex; height:2.176ex;" alt="{\displaystyle \theta =\,}"> </td> <td style="border-style: solid none solid none; text-align: right;"> </td> <td style="border-style: solid none solid none; text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc23e6711d2615a1b719a0965ad958d5252f22e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.064ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccot}(r)}"> </td> <td style="border-style: solid none solid none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"> </td> <td style="border-style: solid none solid none;"> </td> <td style="border-style: solid none solid none; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15954a4d290f0804d541cb1176021ba4c79eb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.543ex; height:2.176ex;" alt="{\displaystyle \pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr></tbody></table> where the first four solutions can be written in expanded form as: <table class="wikitable" style="border: none;"> <caption> </caption> <tbody><tr> <th>Equation</th> <th><a href="/wiki/If_and_only_if" title="If and only if">if and only if</a></th> <th colspan="7">Solution </th></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d80b5d58bed05745127a4e7956b44cb9d9f1e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.587ex; height:2.509ex;" alt="{\displaystyle \sin \theta =y}"> </td> <td style="text-align: center;"><a href="/wiki/Logical_equality" title="Logical equality"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></a> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\;\;\;\,\arcsin(y)+2\pi h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\;\;\;\,\arcsin(y)+2\pi h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b985fbeee57349e5cdbabfe042cc9fa4930256e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.112ex; height:2.843ex;" alt="{\displaystyle \theta =\;\;\;\,\arcsin(y)+2\pi h}">           or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\arcsin(y)+2\pi h+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\arcsin(y)+2\pi h+\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264621c5a1206f15e9daf74a5599887a6efbb831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.157ex; height:2.843ex;" alt="{\displaystyle \theta =-\arcsin(y)+2\pi h+\pi }"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4100e879aee4c465c42987e7ce794d95159a74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786f91f9500bb23dd6f8af17f6bfbbbfe861a880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.606ex; height:2.176ex;" alt="{\displaystyle \csc \theta =r}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(r)+2\pi h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(r)+2\pi h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fd8aa2110c4fd13865c39d9ae1fba93e7e4234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.131ex; height:2.843ex;" alt="{\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(r)+2\pi h}">           or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\operatorname {arccsc}(r)+2\pi h+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\operatorname {arccsc}(r)+2\pi h+\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13edec9a3e7055ab28042dcf97a0dfaaf3ceffe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.176ex; height:2.843ex;" alt="{\displaystyle \theta =-\operatorname {arccsc}(r)+2\pi h+\pi }"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4100e879aee4c465c42987e7ce794d95159a74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\;\;\;\,\arccos(x)+2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\;\;\;\,\arccos(x)+2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1f224030a6a467d1b5fb630a4951bb067492d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.414ex; height:2.843ex;" alt="{\displaystyle \theta =\;\;\;\,\arccos(x)+2\pi k}">          or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\arccos(x)+2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\arccos(x)+2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb79d15150f5c543419c4fe9f7f670dcb8cd37c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.287ex; height:2.843ex;" alt="{\displaystyle \theta =-\arccos(x)+2\pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/025c4b681de7464f3021389a4079c30acf4dba15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.606ex; height:2.176ex;" alt="{\displaystyle \sec \theta =r}"> </td> <td style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(r)+2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(r)+2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150b83bc8a2054969f7d82a75949efa7b6d9863a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.003ex; height:2.843ex;" alt="{\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(r)+2\pi k}">          or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\operatorname {arcsec}(r)+2\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\operatorname {arcsec}(r)+2\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760679713f1129e1ed5d8f626a4f3f6aa5c13644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.876ex; height:2.843ex;" alt="{\displaystyle \theta =-\operatorname {arcsec}(r)+2\pi k}"> </td> <td style="text-align: center; padding-left: 1em; padding-right: 1em;">for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"> </td></tr></tbody></table> For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5478cf46b84f76912d70385a7c4fed0516630368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.658ex; height:2.343ex;" alt="{\displaystyle \cos \theta =-1}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b8cf5bc2d3d7687bfdae1a4f83dfb2dadc8e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.664ex; height:2.843ex;" alt="{\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0829895a7f6a03b7e80aac56c7df6e277fd09cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.249ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} .}"> While if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7da1a97177049a9e0ab7dff21fc4f236ff8918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.402ex; height:2.176ex;" alt="{\displaystyle \sin \theta =\pm 1}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>π</mi> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b00e8bcfd84e71b611e9996fab270434bd5ad9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.231ex; height:3.176ex;" alt="{\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3a59bfa1fddae9c292b0a08f06e0d43a9aa257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.249ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {Z} ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> will be even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b319906b628d010522d1d298a34fd173c82af5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.594ex; height:2.176ex;" alt="{\displaystyle \sin \theta =1}"> and it will be odd if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a753e9e8d2529302725c52cb1e524cf019c8256d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.049ex; height:2.343ex;" alt="{\displaystyle \sin \theta =-1.}"> The equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa10eeb8e4176af122afefb2678607209202bb90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.528ex; height:2.343ex;" alt="{\displaystyle \sec \theta =-1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a1adfa171b65eea5a37f0481617e0f093c051f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.528ex; height:2.176ex;" alt="{\displaystyle \csc \theta =\pm 1}"> have the same solutions as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5478cf46b84f76912d70385a7c4fed0516630368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.658ex; height:2.343ex;" alt="{\displaystyle \cos \theta =-1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =\pm 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =\pm 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb87425a631146bd02fca97ccd2eb2e5254bee92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.049ex; height:2.509ex;" alt="{\displaystyle \sin \theta =\pm 1,}"> respectively. In all equations above except for those just solved (i.e. except for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }">/<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \theta =\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \theta =\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14a1adfa171b65eea5a37f0481617e0f093c051f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.528ex; height:2.176ex;" alt="{\displaystyle \csc \theta =\pm 1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }">/<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa10eeb8e4176af122afefb2678607209202bb90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.528ex; height:2.343ex;" alt="{\displaystyle \sec \theta =-1}">), the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> in the solution's formula is uniquely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> (for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,s,x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,s,x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ddf5bbe1cfaa0df8a09b3aedce67af9e5582d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.184ex; height:2.009ex;" alt="{\displaystyle r,s,x,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">). With the help of <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">integer parity</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Parity</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd </mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b79e4e2a4ff506e920adc85584f676081e79a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.83ex; height:6.176ex;" alt="{\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}}"> it is possible to write a solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> that doesn't involve the "plus or minus" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\pm \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>±</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\pm \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea4e6ce8c7579635e067a0825e7e4c8023ccbcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,\pm \,}"> symbol: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cos\;\theta =x\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mspace width="thickmathspace" /> <mi>θ</mi> <mo>=</mo> <mi>x</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cos\;\theta =x\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2572b70ff426ff4451868527b05c8dedbe6fd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.711ex; height:2.176ex;" alt="{\displaystyle cos\;\theta =x\quad }"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \theta =(-1)^{h}\arccos(x)+\pi h+\pi \operatorname {Parity} (h)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> <mi>Parity</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \theta =(-1)^{h}\arccos(x)+\pi h+\pi \operatorname {Parity} (h)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6c413937c230d711b4961e8552551c0e6f27f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.192ex; height:3.176ex;" alt="{\displaystyle \quad \theta =(-1)^{h}\arccos(x)+\pi h+\pi \operatorname {Parity} (h)\quad }"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8157beb771efc7b880c995f7ac38493547153b76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.377ex; height:2.176ex;" alt="{\displaystyle h\in \mathbb {Z} .}"></dd></dl> And similarly for the secant function, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sec\;\theta =r\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>e</mi> <mi>c</mi> <mspace width="thickmathspace" /> <mi>θ</mi> <mo>=</mo> <mi>r</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sec\;\theta =r\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366a3c1b4b4cc3607a6fed38284b9d1268b7d0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.386ex; height:2.176ex;" alt="{\displaystyle sec\;\theta =r\quad }"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \theta =(-1)^{h}\operatorname {arcsec}(r)+\pi h+\pi \operatorname {Parity} (h)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> <mi>Parity</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \theta =(-1)^{h}\operatorname {arcsec}(r)+\pi h+\pi \operatorname {Parity} (h)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3bd7c8c810e6a3faf8e86a01b0e485dd99cb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.781ex; height:3.176ex;" alt="{\displaystyle \quad \theta =(-1)^{h}\operatorname {arcsec}(r)+\pi h+\pi \operatorname {Parity} (h)\quad }"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc76f0267891682a45e2805b691ddd2c26770cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.377ex; height:2.509ex;" alt="{\displaystyle h\in \mathbb {Z} ,}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi h+\pi \operatorname {Parity} (h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> <mi>Parity</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi h+\pi \operatorname {Parity} (h)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d82a187500e2fc0cd3d1ae5d56455f2091d4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.815ex; height:2.843ex;" alt="{\displaystyle \pi h+\pi \operatorname {Parity} (h)}"> equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0845b6872dd6e48f3dc008958cd5d2a07b1490e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.671ex; height:2.176ex;" alt="{\displaystyle \pi h}"> when the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> is even, and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi h+\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi h+\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd19c353e5b0ddde4a5bdc46f72dcc9091bac8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.843ex; height:2.343ex;" alt="{\displaystyle \pi h+\pi }"> when it's odd. <div class="mw-heading mw-heading4"><h4 id="Detailed_example_and_explanation_of_the_"plus_or_minus"_symbol_±">Detailed example and explanation of the "plus or minus" symbol ±</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=6" title="Edit section: Detailed example and explanation of the "plus or minus" symbol ±">edit</a>]</div> The solutions to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c381e360f360137c1bb81d5855dc7a121435d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.887ex; height:2.176ex;" alt="{\displaystyle \sec \theta =x}"> involve the "plus or minus" symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\pm ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>±</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\pm ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037cdd9cc2b7f75f0a96b6d5e662dc3d013a9928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.229ex; height:2.509ex;" alt="{\displaystyle \,\pm ,\,}"> whose meaning is now clarified. Only the solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> will be discussed since the discussion for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c381e360f360137c1bb81d5855dc7a121435d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.887ex; height:2.176ex;" alt="{\displaystyle \sec \theta =x}"> is the same. We are given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07d30937eb8028698054698012f2d76caad7fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1\leq x\leq 1}"> and we know that there is an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> in some interval that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5e6a4b3c4f019578499c98c86d6f78bb65186d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.664ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x.}"> We want to find this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082e8402f1cbddec479b88e2ff0d1be5e9b95bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.737ex; height:2.176ex;" alt="{\displaystyle \theta .}"> The table above indicates that the solution is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89965cb1eb83d4221f9c222f506fbda20257db8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:39.138ex; height:2.343ex;" alt="{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }"> which is a shorthand way of saying that (at least) one of the following statement is true: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\theta =\arccos x+2\pi k\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\theta =\arccos x+2\pi k\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fd9977ac4627a8692209059a856152f09d9038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.444ex; height:2.343ex;" alt="{\displaystyle \,\theta =\arccos x+2\pi k\,}"> for some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e185ab9c990830d5055fa3ae698a4225ce67e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle k,}"> or</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\theta =-\arccos x+2\pi k\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\theta =-\arccos x+2\pi k\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e90344cefecff1959bb231d04eb3ed37d88f6a31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.639ex; height:2.343ex;" alt="{\displaystyle \,\theta =-\arccos x+2\pi k\,}"> for some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb6778a29f576eb23da1dbddffb73b2571359ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle k.}"></li></ol> As mentioned above, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x=\pi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x=\pi \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fb9d72b145c1521dcd0a63c55120992a613953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.139ex; height:1.676ex;" alt="{\displaystyle \,\arccos x=\pi \,}"> (which by definition only happens when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\cos \pi =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>π</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\cos \pi =-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/694e678d898c26593d2ac9a3fc4c8d5b808fadf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.327ex; height:2.343ex;" alt="{\displaystyle x=\cos \pi =-1}">) then both statements (1) and (2) hold, although with different values for the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is the integer from statement (1), meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi +2\pi K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi +2\pi K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55babcbc112a957f4b7a9aba7788a8911648c867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.922ex; height:2.343ex;" alt="{\displaystyle \theta =\pi +2\pi K}"> holds, then the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> for statement (2) is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3c16df6a96cbc6bb45ef18a401e161c3dd348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.069ex; height:2.343ex;" alt="{\displaystyle K+1}"> (because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\pi +2\pi (1+K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\pi +2\pi (1+K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfa0ea3e184c4a34858dd8ad15d748045d73ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.542ex; height:2.843ex;" alt="{\displaystyle \theta =-\pi +2\pi (1+K)}">). However, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a83e8331c458b7378b08c5ef4cce193113e68e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.399ex; height:2.676ex;" alt="{\displaystyle x\neq -1}"> then the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is unique and completely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082e8402f1cbddec479b88e2ff0d1be5e9b95bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.737ex; height:2.176ex;" alt="{\displaystyle \theta .}"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x=0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5285501f3c71ef950c0d7c5c98409272b4091af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.969ex; height:2.176ex;" alt="{\displaystyle \,\arccos x=0\,}"> (which by definition only happens when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\cos 0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\cos 0=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b41b0c72e4a4c5e629ac92abdfebcdf8d9c750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.35ex; height:2.176ex;" alt="{\displaystyle x=\cos 0=1}">) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\pm \arccos x=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\pm \arccos x=0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4adaf4b8aaf1fbfd568698bbb9c8b943422dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.165ex; height:2.176ex;" alt="{\displaystyle \,\pm \arccos x=0\,}"> (because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\arccos x=+0=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\arccos x=+0=0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba64bae7fe364948569f069dac4dc32384c339e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.234ex; height:2.343ex;" alt="{\displaystyle \,+\arccos x=+0=0\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,-\arccos x=-0=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,-\arccos x=-0=0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/373b3858fe8d4b6ede015a7c4027416cd9b33634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.234ex; height:2.343ex;" alt="{\displaystyle \,-\arccos x=-0=0\,}"> so in both cases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\pm \arccos x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\pm \arccos x\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc02e14165d5d4081e31496235e037051463ebae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.904ex; height:2.176ex;" alt="{\displaystyle \,\pm \arccos x\,}"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x=0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5285501f3c71ef950c0d7c5c98409272b4091af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.969ex; height:2.176ex;" alt="{\displaystyle \,\arccos x=0\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x=\pi ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x=\pi ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e32e5a5f499793e39913d4911b1a255bc3b7f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.786ex; height:2.009ex;" alt="{\displaystyle \,\arccos x=\pi ,\,}"> we now focus on the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x\neq 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x\neq 0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493814b3fe93606432bf26cc92e714136de9550d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.969ex; height:2.676ex;" alt="{\displaystyle \,\arccos x\neq 0\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x\neq \pi ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>≠</mo> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x\neq \pi ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf195cd203502fe6e289028d621c83559df81934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.786ex; height:2.676ex;" alt="{\displaystyle \,\arccos x\neq \pi ,\,}"> So assume this from now on. The solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> is still <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89965cb1eb83d4221f9c222f506fbda20257db8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:39.138ex; height:2.343ex;" alt="{\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }"> which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\arccos x\neq 0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\arccos x\neq 0\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493814b3fe93606432bf26cc92e714136de9550d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.969ex; height:2.676ex;" alt="{\displaystyle \,\arccos x\neq 0\,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,0<\arccos x<\pi ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mn>0</mn> <mo><</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,0<\arccos x<\pi ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c03825ece2c8a876156a21f73be42871542cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.047ex; height:2.509ex;" alt="{\displaystyle \,0<\arccos x<\pi ,\,}"> statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is needed to determine which one holds. For example, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"> and that all that is known about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,-\pi \leq \theta \leq \pi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>−</mo> <mi>π</mi> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,-\pi \leq \theta \leq \pi \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ecee598b5ccbad923d5009287c17e882d6a05c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.534ex; height:2.343ex;" alt="{\displaystyle \,-\pi \leq \theta \leq \pi \,}"> (and nothing more is known). Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490c0cc05c39c914f77aca71be5bc3aa4b621628" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.067ex; height:4.676ex;" alt="{\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}}"> and moreover, in this particular case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}"> (for both the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}"> case and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,-\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,-\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e22e29cfe7cbcbdf5fff213a4e77bcd52d18d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,-\,}"> case) and so consequently, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo>±</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo>±</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bf4f3cc6db8e7b588e7d5abc8cc61bedfc3ab0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.937ex; height:4.843ex;" alt="{\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.}"> This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> could be either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\pi /2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\pi /2\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d82c65bde7c9d91d558abb41b39b0ce2a6a0bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle \,\pi /2\,}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,-\pi /2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,-\pi /2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00d06b91126d9dfbca974f3aed8d397868caf93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle \,-\pi /2.}"> Without additional information it is not possible to determine which of these values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> has. An example of some additional information that could determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> would be knowing that the angle is above the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-axis (in which case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc4628b0f731f81bfabcd8edaca00aa186f03bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.846ex; height:2.843ex;" alt="{\displaystyle \theta =\pi /2}">) or alternatively, knowing that it is below the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">-axis (in which case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-\pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20db6caa2c3cdfc98f3ae32a257da1fefffbdccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \theta =-\pi /2}">). <div class="mw-heading mw-heading4"><h4 id="Equal_identical_trigonometric_functions">Equal identical trigonometric functions</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=7" title="Edit section: Equal identical trigonometric functions">edit</a>]</div> The table below shows how two angles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> must be related if their values under a given trigonometric function are equal or negatives of each other. <table class="wikitable" style="text-align: center; ;"> <caption> </caption> <tbody><tr> <th scope="col">Equation </th> <th scope="col"><a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> </th> <th scope="col">Solution (for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">) </th> <th scope="col">Also a solution to </th></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\sin \theta =\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\sin \theta =\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a032bec6308298192f93bb6a635a5604578ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.39ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\sin \theta =\sin \varphi }"> </th> <td><a href="/wiki/Logical_equality" title="Logical equality"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"></a> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mspace width="1em" /> </mphantom> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>2</mn> </mphantom> </mrow> </mrow> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae11acf5cfa7c6dfd877b46f625474a32b759a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.586ex; height:3.176ex;" alt="{\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\csc \theta =\csc \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\csc \theta =\csc \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93eb9d09db3260359f17ad16c64ca1f467681c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.641ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\csc \theta =\csc \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\cos \theta =\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\cos \theta =\cos \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37967ff588a6d3b7545541ffc1072a7fd1ad3f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.901ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\cos \theta =\cos \varphi }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> </mphantom> </mrow> </mrow> <mo>±</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aecfcb615c0fe71c5d9c5dfe441d55b61bb97452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.529ex; height:2.676ex;" alt="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\sec \theta =\sec \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\sec \theta =\sec \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8006dec7ccb69458b801fd439b23e70a40541ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.641ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\sec \theta =\sec \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\tan \theta =\tan \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\tan \theta =\tan \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75833cde22b59ed3fc3f517b3b042ba84e72ea6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.398ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\tan \theta =\tan \varphi }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mphantom> </mrow> </mrow> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>2</mn> </mphantom> </mrow> </mrow> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6008f51d3d8fe282ce123dabc271ad23e1c457ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.364ex; height:3.176ex;" alt="{\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\cot \theta =\cot \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\cot \theta =\cot \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c444736f71207dc62d4219abb03eddc4766a463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.877ex; height:2.676ex;" alt="{\displaystyle {\phantom {-}}\cot \theta =\cot \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sin \theta =\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sin \theta =\sin \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7e1c1417c740054dd2bc179dbf28aae5867425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.39ex; height:2.676ex;" alt="{\displaystyle -\sin \theta =\sin \varphi }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>2</mn> </mphantom> </mrow> </mrow> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06a910355008e2582bab5a51bc248607949fc94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.364ex; height:3.176ex;" alt="{\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\csc \theta =\csc \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>csc</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\csc \theta =\csc \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a97eb1fc6d71683a2cf106381a4a51d8e6fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.641ex; height:2.676ex;" alt="{\displaystyle -\csc \theta =\csc \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cos \theta =\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cos \theta =\cos \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4674608e08e2bad8845533b9a848bcc127e76fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.901ex; height:2.676ex;" alt="{\displaystyle -\cos \theta =\cos \varphi }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> </mphantom> </mrow> </mrow> <mo>±</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d78b7ac9fcd3445cc945bb80d6f781fae4292d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.701ex; height:2.676ex;" alt="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sec \theta =\sec \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sec \theta =\sec \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a067c013966497b206ed9aeb558ea28ac75046b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.641ex; height:2.676ex;" alt="{\displaystyle -\sec \theta =\sec \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\tan \theta =\tan \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\tan \theta =\tan \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a469909fc652802c9cf89b7d6e25df93ed15eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.398ex; height:2.676ex;" alt="{\displaystyle -\tan \theta =\tan \varphi }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> </mphantom> </mrow> </mrow> <mo>−</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>2</mn> </mphantom> </mrow> </mrow> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/818ba398f9812c9af4440c1aadac2fa756f5985e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.529ex; height:2.676ex;" alt="{\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cot \theta =\cot \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cot \theta =\cot \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071458975b9fde73f6f3da55c2160819e54ad2b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.877ex; height:2.676ex;" alt="{\displaystyle -\cot \theta =\cot \varphi }"> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">⇕</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a4e0126adf220b309b61344aeed37f3f69555e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.24ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\&\Updownarrow \\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iff }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iff }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff942842a50b24e7585cc42c5b50c34650e3aa99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:1.843ex;" alt="{\displaystyle \iff }"> </td> <td style="text-align: left; padding-left: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> </mphantom> </mrow> </mrow> <mo>±</mo> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mn>2</mn> </mphantom> </mrow> </mrow> <mi>π</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>+</mo> <mi>π</mi> </mphantom> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34378e538f28c90b43c8748eb3db5898c6913d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.529ex; height:2.676ex;" alt="{\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mrow> <mo>|</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>csc</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d2b973b8facbab50b42f770a3210cbeb1fc3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:18.737ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}"> </td></tr></tbody></table> The vertical double arrow <a href="/wiki/Logical_equality" title="Logical equality"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Updownarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Updownarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2dfaf6c3bec677ed3666d6cd91648833837c620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.42ex; height:2.843ex;" alt="{\displaystyle \Updownarrow }"></a> in the last row indicates that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b288bae55679a588073a5dbdf9a9dd3e1ba987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.782ex; height:2.843ex;" alt="{\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|}"> if and only if they satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac101a27f4c7163098010f5c4c282af3e4d9e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.327ex; height:2.843ex;" alt="{\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.}"> <dl><dt>Set of all solutions to elementary trigonometric equations</dt></dl> Thus given a single solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> to an elementary trigonometric equation (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta =y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d80b5d58bed05745127a4e7956b44cb9d9f1e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.587ex; height:2.509ex;" alt="{\displaystyle \sin \theta =y}"> is such an equation, for instance, and because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arcsin y)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arcsin y)=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28657df7af060ab137772fb383921637b8685563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.424ex; height:2.843ex;" alt="{\displaystyle \sin(\arcsin y)=y}"> always holds, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta :=\arcsin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>:=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta :=\arcsin y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b696838e8ecd496b183b546b6f4c098e361940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.34ex; height:2.509ex;" alt="{\displaystyle \theta :=\arcsin y}"> is always a solution), the set of all solutions to it are: <table class="wikitable" style="border: none;"> <caption> </caption> <tbody><tr> <th>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> solves</th> <th>then</th> <th colspan="7">Set of all solutions (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }">) </th></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sin \theta =y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sin \theta =y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a16d8149cd24fe0a64f8b5a9952d1c06d6ddea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.232ex; height:2.509ex;" alt="{\displaystyle \;\sin \theta =y}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\sin \varphi =y\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\sin \varphi =y\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29855657a5cc695377b0c68659a3f499b03ce794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.639ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\sin \varphi =y\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48662a8fbdcef50566a70d7b30a8435d99ecf3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.995ex; height:2.843ex;" alt="{\displaystyle (\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d50dbac0115ffc2f04fb66945709e8fb307f6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.745ex; height:2.343ex;" alt="{\displaystyle \,+\,2}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9cc2e0380c9e95020377114074b04c74275376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.787ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} )}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cup \,(-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∪</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cup \,(-\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84c7006a7d279358d0374aacf6991e358e7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle \,\cup \,(-\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2359073fe90a84a705e02f0c1e63b32df850a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle -\pi }"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2\pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2\pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25546bc1c0803d7167c5e92e838f52672002406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="{\displaystyle +2\pi \mathbb {Z} )}"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\csc \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\csc \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85be4e153bc6429c06f40f7849c4573999902d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.251ex; height:2.176ex;" alt="{\displaystyle \;\csc \theta =r}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\csc \varphi =r\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>csc</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\csc \varphi =r\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83d305cb361219fb6cdc141447b69e6b12d4d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.658ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\csc \varphi =r\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48662a8fbdcef50566a70d7b30a8435d99ecf3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.995ex; height:2.843ex;" alt="{\displaystyle (\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d50dbac0115ffc2f04fb66945709e8fb307f6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.745ex; height:2.343ex;" alt="{\displaystyle \,+\,2}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9cc2e0380c9e95020377114074b04c74275376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.787ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} )}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cup \,(-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∪</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cup \,(-\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84c7006a7d279358d0374aacf6991e358e7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle \,\cup \,(-\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2359073fe90a84a705e02f0c1e63b32df850a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle -\pi }"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2\pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2\pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25546bc1c0803d7167c5e92e838f52672002406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="{\displaystyle +2\pi \mathbb {Z} )}"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9010e4e04661c22ec8f5084ec9081cb8444434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.662ex; height:2.176ex;" alt="{\displaystyle \;\cos \theta =x}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\cos \varphi =x\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\cos \varphi =x\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56655c38c8de08659511d2c8628d27a7b8d79259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.069ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\cos \varphi =x\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48662a8fbdcef50566a70d7b30a8435d99ecf3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.995ex; height:2.843ex;" alt="{\displaystyle (\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d50dbac0115ffc2f04fb66945709e8fb307f6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.745ex; height:2.343ex;" alt="{\displaystyle \,+\,2}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9cc2e0380c9e95020377114074b04c74275376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.787ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} )}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cup \,(-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∪</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cup \,(-\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84c7006a7d279358d0374aacf6991e358e7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle \,\cup \,(-\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2\pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2\pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25546bc1c0803d7167c5e92e838f52672002406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="{\displaystyle +2\pi \mathbb {Z} )}"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sec \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sec \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43fda4922fe67e1b336b1633aeb88a676745700d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.251ex; height:2.176ex;" alt="{\displaystyle \;\sec \theta =r}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\sec \varphi =r\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\sec \varphi =r\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7b2dffc5eac7913e9329c6c4dbd36c4ebae84c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.658ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\sec \varphi =r\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48662a8fbdcef50566a70d7b30a8435d99ecf3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.995ex; height:2.843ex;" alt="{\displaystyle (\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d50dbac0115ffc2f04fb66945709e8fb307f6fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.745ex; height:2.343ex;" alt="{\displaystyle \,+\,2}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9cc2e0380c9e95020377114074b04c74275376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.787ex; height:2.843ex;" alt="{\displaystyle \pi \mathbb {Z} )}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cup \,(-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>∪</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cup \,(-\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84c7006a7d279358d0374aacf6991e358e7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle \,\cup \,(-\theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +2\pi \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +2\pi \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b25546bc1c0803d7167c5e92e838f52672002406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="{\displaystyle +2\pi \mathbb {Z} )}"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\tan \theta =s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\tan \theta =s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f7b4a378cafd5d26673ca2ff28f815ce3ee5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.671ex; height:2.176ex;" alt="{\displaystyle \;\tan \theta =s}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\tan \varphi =s\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\tan \varphi =s\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ac1e8a0176a5b2b466b4cd50fc12f7ad603778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.078ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\tan \varphi =s\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee8eb7deec783f26ce796c775792314e2896763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.882ex; height:2.176ex;" alt="{\displaystyle \pi \mathbb {Z} }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"> </td></tr> <tr> <td style="text-align: center; padding: 0.5% 2em 0.5% 2em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\cot \theta =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\cot \theta =r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2976955c21d26000b12c332e3f41bd6a3e64859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.369ex; height:2.176ex;" alt="{\displaystyle \;\cot \theta =r}"> </td> <td style="text-align: center;">then </td> <td style="border-style: solid none solid none; text-align: left; padding-left: 1em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi :\cot \varphi =r\}=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>:</mo> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\varphi :\cot \varphi =r\}=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86dfca4fd6581444d9b4bb794eae2ed05dd903b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.776ex; height:2.843ex;" alt="{\displaystyle \{\varphi :\cot \varphi =r\}=\,}"> </td> <td style="border-style: solid none solid none; text-align: right; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,+\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,+\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.582ex; height:2.176ex;" alt="{\displaystyle \,+\,}"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee8eb7deec783f26ce796c775792314e2896763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.882ex; height:2.176ex;" alt="{\displaystyle \pi \mathbb {Z} }"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid none solid none; text-align: left; padding: 0;"> </td> <td style="border-style: solid solid solid none; text-align: left; padding-left: 0; padding-right: 2em;"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Transforming_equations">Transforming equations</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=8" title="Edit section: Transforming equations">edit</a>]</div> The equations above can be transformed by using the reflection and shift identities:<a href="#cite_note-13">[12]</a> <table class="wikitable" style="text-align: center;"> <caption>Transforming equations by shifts and reflections </caption> <tbody><tr> <th scope="col">Argument: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\;~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\;~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd657f7ea0d05bca4f04f7d9782b032f85c2a09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:7.23ex; height:2.509ex;" alt="{\displaystyle {\underline {\;~~~~~~\;}}=}"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d715f4b31b39f52768eba6405b658be5dc573f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.899ex; height:2.343ex;" alt="{\displaystyle -\theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}\pm \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}\pm \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97488141e235bbaa57eec86e761726db9b02cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.099ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}\pm \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \pm \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>±</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \pm \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3995ad08794a60300c164ca789cd7ead7dd29d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.263ex; height:2.176ex;" alt="{\displaystyle \pi \pm \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3\pi }{2}}\pm \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\pi }{2}}\pm \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55baea0696a2c6e4979ce5f76f9b4392da36d358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.261ex; height:5.176ex;" alt="{\displaystyle {\frac {3\pi }{2}}\pm \theta }"> </th> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k\pi \pm \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo>±</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2k\pi \pm \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53cfafa381ede5c10b75f56569329d2c94a86eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.283ex; height:2.509ex;" alt="{\displaystyle 2k\pi \pm \theta ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k\in \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k\in \mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9010c6a9903487b97e34c72404f3c91f828f7477" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.412ex; height:2.843ex;" alt="{\displaystyle (k\in \mathbb {Z} )}"> </th></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ceb10e885d37eeff88587ccd5b372792054ab5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.118ex; height:3.176ex;" alt="{\displaystyle \sin {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be85a6ba1cec5a97446d96b01fff17c347dda051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.528ex; height:2.343ex;" alt="{\displaystyle -\sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a345d73492d88e0f3a89adf2a75d0c7b0159347a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.784ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fed014d20d681cdaee606f27505e257ee725991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.509ex;" alt="{\displaystyle \mp \sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473ee6c118fd4db3f126fbef303d3340aa27b3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.784ex; height:2.343ex;" alt="{\displaystyle -\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067805ec05d3058f85084ab61533774682317704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.528ex; height:2.176ex;" alt="{\displaystyle \pm \sin \theta }"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcae5dff8628a8c605a893bc8600ac418d77862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.243ex; height:2.676ex;" alt="{\displaystyle \csc {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3fab7ba098d1a46fe489cbacfd9ca6de61bd58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle -\csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563e2456da645d5729bafc2f581f30d798f6561e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c45af4cb34db05f6c1232905244f7ee611cc8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.654ex; height:2.509ex;" alt="{\displaystyle \mp \csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049e51921dba06c7bdb1aa8f5b8e3e1b516ac0aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle -\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54148f5e83d64c68d824d7c9e5ab7d340beacaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.654ex; height:2.176ex;" alt="{\displaystyle \pm \csc \theta }"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a358e8af7eb5ab63d3bd0f0b5cf3dfc12877b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.373ex; height:2.676ex;" alt="{\displaystyle \cos {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a345d73492d88e0f3a89adf2a75d0c7b0159347a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.784ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fed014d20d681cdaee606f27505e257ee725991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.528ex; height:2.509ex;" alt="{\displaystyle \mp \sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473ee6c118fd4db3f126fbef303d3340aa27b3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.784ex; height:2.343ex;" alt="{\displaystyle -\cos \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067805ec05d3058f85084ab61533774682317704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.528ex; height:2.176ex;" alt="{\displaystyle \pm \sin \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\cos \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a345d73492d88e0f3a89adf2a75d0c7b0159347a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.784ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\cos \theta }"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e26aec387abad74433acd2f412ede8a9d6a33a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.243ex; height:2.676ex;" alt="{\displaystyle \sec {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563e2456da645d5729bafc2f581f30d798f6561e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c45af4cb34db05f6c1232905244f7ee611cc8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.654ex; height:2.509ex;" alt="{\displaystyle \mp \csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049e51921dba06c7bdb1aa8f5b8e3e1b516ac0aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle -\sec \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \csc \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>csc</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \csc \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54148f5e83d64c68d824d7c9e5ab7d340beacaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.654ex; height:2.176ex;" alt="{\displaystyle \pm \csc \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\phantom {-}}\sec \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sec</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\phantom {-}}\sec \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563e2456da645d5729bafc2f581f30d798f6561e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.654ex; height:2.343ex;" alt="{\displaystyle {\phantom {-}}\sec \theta }"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa00c13b435c7c29bf0233c8c22506ed3e1190f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.622ex; height:3.009ex;" alt="{\displaystyle \tan {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc602e43e7db125fc42bf760bc241d70fc66e00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.032ex; height:2.343ex;" alt="{\displaystyle -\tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28cdbc3db46ac93c56a835a6888aaa90c80b1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.772ex; height:2.509ex;" alt="{\displaystyle \mp \cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eba069efdda8d9fbab407909b7cd87c72e187c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.032ex; height:2.176ex;" alt="{\displaystyle \pm \tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28cdbc3db46ac93c56a835a6888aaa90c80b1b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.772ex; height:2.509ex;" alt="{\displaystyle \mp \cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eba069efdda8d9fbab407909b7cd87c72e187c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.032ex; height:2.176ex;" alt="{\displaystyle \pm \tan \theta }"> </td></tr> <tr> <th scope="row"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot {\underline {\;~~~~~~~~~~~~~~\;}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mspace width="thickmathspace" /> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mspace width="thickmathspace" /> </mrow> <mo>_</mo> </munder> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot {\underline {\;~~~~~~~~~~~~~~\;}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d8c9bca42c55454b6f1230fc5510fda1d95048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.509ex; margin-bottom: -0.829ex; width:15.362ex; height:3.009ex;" alt="{\displaystyle \cot {\underline {\;~~~~~~~~~~~~~~\;}}=}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2ab9bdbbf4c0cdf134ac834f76fc8a2c2da811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.772ex; height:2.343ex;" alt="{\displaystyle -\cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b33a68faba46ae8d8130f59ac567ce583957bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle \mp \tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a39fdb6123e364b0bab2b956d40243b883c46a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.772ex; height:2.176ex;" alt="{\displaystyle \pm \cot \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mp \tan \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mp \tan \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b33a68faba46ae8d8130f59ac567ce583957bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.032ex; height:2.509ex;" alt="{\displaystyle \mp \tan \theta }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \cot \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \cot \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a39fdb6123e364b0bab2b956d40243b883c46a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.772ex; height:2.176ex;" alt="{\displaystyle \pm \cot \theta }"> </td></tr></tbody></table> These formulas imply, in particular, that the following hold: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt 0.3em 0.429em 0.3em 0.3em 0.429em 0.3em 0.3em 0.429em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>+</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mo>−</mo> </mphantom> </mrow> </mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316baee0254e64a99779ac4721054f23295aeec6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -20.935ex; margin-bottom: -0.236ex; width:73.949ex; height:43.509ex;" alt="{\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}"> where swapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \leftrightarrow \csc ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo stretchy="false">↔</mo> <mi>csc</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \leftrightarrow \csc ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444ebe59dce7921b3b9c091e1c090b1ba94dcb0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.098ex; height:2.509ex;" alt="{\displaystyle \sin \leftrightarrow \csc ,}"> swapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \leftrightarrow \sec ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo stretchy="false">↔</mo> <mi>sec</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \leftrightarrow \sec ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f11d131b8807f72bd310ce9929b9809aa21212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.353ex; height:2.176ex;" alt="{\displaystyle \cos \leftrightarrow \sec ,}"> and swapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \leftrightarrow \cot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo stretchy="false">↔</mo> <mi>cot</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \leftrightarrow \cot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1457f43924ff66f57a63793bab71bc3e9e09920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.073ex; height:2.009ex;" alt="{\displaystyle \tan \leftrightarrow \cot }"> gives the analogous equations for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc ,\sec ,{\text{ and }}\cot ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>,</mo> <mi>sec</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>cot</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc ,\sec ,{\text{ and }}\cot ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f826ee0ef9bf1d86b7f170c8f7eeea594a6e4e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.072ex; height:2.509ex;" alt="{\displaystyle \csc ,\sec ,{\text{ and }}\cot ,}"> respectively. So for example, by using the equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b652c8d249264870ff7be7c6524d60569ed1910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.028ex; height:3.343ex;" alt="{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,}"> the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta =x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f628c27d546b187d68f7ec5a5caea7e619b8742b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.017ex; height:2.176ex;" alt="{\displaystyle \cos \theta =x}"> can be transformed into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff5d99bbd07a8ade3b2b3f363b659032356066ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.769ex; height:3.343ex;" alt="{\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,}"> which allows for the solution to the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sin \varphi =x\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sin \varphi =x\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6bd46b040b356fee3f42d86559de774a04305a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.481ex; height:2.676ex;" alt="{\displaystyle \;\sin \varphi =x\;}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi :={\frac {\pi }{2}}-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi :={\frac {\pi }{2}}-\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8ceadd294f6037b0feccae4892956101507b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.974ex; height:3.176ex;" alt="{\textstyle \varphi :={\frac {\pi }{2}}-\theta }">) to be used; that solution being: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>k</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e32b11e0cf4752d653eb28c3a3e56272a3e89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.828ex; height:3.176ex;" alt="{\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,}"> which becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>k</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ba4cdd1fc682732c9fbc35460c67432efc4c44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.599ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }"> where using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{k}=(-1)^{-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{k}=(-1)^{-k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffd0aad266d18c029225c916f43e4e8153520b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.114ex; height:3.176ex;" alt="{\displaystyle (-1)^{k}=(-1)^{-k}}"> and substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:=-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:=</mo> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:=-k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598e75c0432a261a06b839c4fd831770d1d669c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.104ex; height:2.343ex;" alt="{\displaystyle h:=-k}"> proves that another solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\cos \theta =x\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\cos \theta =x\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59c7e3d10682f7869c6d9909766f49071bc24618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.307ex; height:2.176ex;" alt="{\displaystyle \;\cos \theta =x\;}"> is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>h</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86ec9a49d6a9d38aab9cfebcf17188956b802061" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.692ex; height:4.676ex;" alt="{\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .}"> The substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6635da62b87bb172a6e5a798c626fbe2ca4fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.011ex; height:4.676ex;" alt="{\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;}"> may be used express the right hand side of the above formula in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\arccos x\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\arccos x\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade872dceed7c090547fa7c1694fe0ea21393600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.225ex; height:1.676ex;" alt="{\displaystyle \;\arccos x\;}"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\arcsin x.\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\arcsin x.\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370f860d40bc6d0a244a7266c123f36f778ce54d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.616ex; height:2.176ex;" alt="{\displaystyle \;\arcsin x.\;}"> <div class="mw-heading mw-heading3"><h3 id="Relationships_between_trigonometric_functions_and_inverse_trigonometric_functions">Relationships between trigonometric functions and inverse trigonometric functions</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=9" title="Edit section: Relationships between trigonometric functions and inverse trigonometric functions">edit</a>]</div> Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"> then applying the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is positive, and thus the result has to be corrected through the use of <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a> and the <a href="/wiki/Sign_function" title="Sign function">signum</a> (sgn) operation. <table class="wikitable"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acafc444aea85d63a40dabf84f035a6b4955a948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.755ex; height:2.843ex;" alt="{\displaystyle \sin(\theta )}"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaac7b75cda6d5570780075aa2622d27b21117cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.011ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )}"> </th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbdbf63a30c035f2e8232b47930e745680b6e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \tan(\theta )}"> </th> <th>Diagram </th></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f111f8c1e5de2f92ee17eeabd64c4ea1bcd55196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle \arcsin(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arcsin(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arcsin(x))=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f91ccd428ab0d060b0864c8b06f3b5d191da98c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.194ex; height:2.843ex;" alt="{\displaystyle \sin(\arcsin(x))=x}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b7e76b5f09150c7d53eefd98da99a7c6003b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.83ex; height:3.509ex;" alt="{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b893b2464d6972d04108e36baf9bba7a1b750a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.915ex; height:6.009ex;" alt="{\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Trigonometric_functions_and_inverse3.svg/150px-Trigonometric_functions_and_inverse3.svg.png" decoding="async" width="150" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Trigonometric_functions_and_inverse3.svg/225px-Trigonometric_functions_and_inverse3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Trigonometric_functions_and_inverse3.svg/300px-Trigonometric_functions_and_inverse3.svg.png 2x" data-file-width="298" data-file-height="207" /></a> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa15507445b7175a6143e9d16ba98a9849374c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.357ex; height:2.843ex;" alt="{\displaystyle \arccos(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96715129befa92fe44fde97e001a8bed280dcb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.83ex; height:3.509ex;" alt="{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arccos(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arccos(x))=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0545e90ee0aaf7439a95c5f9b39f83bb70294b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.705ex; height:2.843ex;" alt="{\displaystyle \cos(\arccos(x))=x}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e712485c56ecce77ce7d7f2da6426b7fb129db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.171ex; height:6.176ex;" alt="{\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Trigonometric_functions_and_inverse.svg/150px-Trigonometric_functions_and_inverse.svg.png" decoding="async" width="150" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Trigonometric_functions_and_inverse.svg/225px-Trigonometric_functions_and_inverse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Trigonometric_functions_and_inverse.svg/300px-Trigonometric_functions_and_inverse.svg.png 2x" data-file-width="363" data-file-height="186" /></a> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2ee2a040ba3b62484ca1518869f51e2c3e5e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.605ex; height:2.843ex;" alt="{\displaystyle \arctan(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a32bc142eb5da5fe334aa7c496e99fc67ff33f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.915ex; height:6.009ex;" alt="{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a657dfc8a2af1dc1a57a7ea86536226babe99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.171ex; height:6.509ex;" alt="{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\arctan(x))=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\arctan(x))=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee0b18e03bd0f966fa61ace657efc626c178a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.202ex; height:2.843ex;" alt="{\displaystyle \tan(\arctan(x))=x}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Trigonometric_functions_and_inverse2.svg/150px-Trigonometric_functions_and_inverse2.svg.png" decoding="async" width="150" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Trigonometric_functions_and_inverse2.svg/225px-Trigonometric_functions_and_inverse2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Trigonometric_functions_and_inverse2.svg/300px-Trigonometric_functions_and_inverse2.svg.png 2x" data-file-width="296" data-file-height="195" /></a> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3e88b0c04d1517ce13a8d20b4303dd2b4a0cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.345ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccot}(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\operatorname {arccot}(x))={\frac {1}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\operatorname {arccot}(x))={\frac {1}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/439e7bea899c5cbd8a618ef9fa6fda3876ea21e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.655ex; height:6.509ex;" alt="{\displaystyle \sin(\operatorname {arccot}(x))={\frac {1}{\sqrt {1+x^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\operatorname {arccot}(x))={\frac {x}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\operatorname {arccot}(x))={\frac {x}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d284151fab701b7202fc1d8baad5ec1d4e408a5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.91ex; height:6.009ex;" alt="{\displaystyle \cos(\operatorname {arccot}(x))={\frac {x}{\sqrt {1+x^{2}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\operatorname {arccot}(x))={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arccot}(x))={\frac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3454ed4d43e723fd550452a6c9197a27348ffca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.778ex; height:5.176ex;" alt="{\displaystyle \tan(\operatorname {arccot}(x))={\frac {1}{x}}}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Trigonometric_functions_and_inverse4.svg/150px-Trigonometric_functions_and_inverse4.svg.png" decoding="async" width="150" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Trigonometric_functions_and_inverse4.svg/225px-Trigonometric_functions_and_inverse4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Trigonometric_functions_and_inverse4.svg/300px-Trigonometric_functions_and_inverse4.svg.png 2x" data-file-width="297" data-file-height="195" /></a> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c3e0ed55bb9513d00d5949d741eb4ffcd77d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.226ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\operatorname {arcsec}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\operatorname {arcsec}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560baae36825a6fb103b98a6ac2fad3a71892117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.537ex; height:7.009ex;" alt="{\displaystyle \sin(\operatorname {arcsec}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\operatorname {arcsec}(x))={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\operatorname {arcsec}(x))={\frac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdff1552bb1555be07cab83a7d5a32053f8d255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.411ex; height:5.176ex;" alt="{\displaystyle \cos(\operatorname {arcsec}(x))={\frac {1}{x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\operatorname {arcsec}(x))=\operatorname {sgn}(x){\sqrt {x^{2}-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arcsec}(x))=\operatorname {sgn}(x){\sqrt {x^{2}-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7b27c40361d9d3ecada5f3a4f771fbbbe41c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.715ex; height:3.509ex;" alt="{\displaystyle \tan(\operatorname {arcsec}(x))=\operatorname {sgn}(x){\sqrt {x^{2}-1}}}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Trigonometric_functions_and_inverse6.svg/150px-Trigonometric_functions_and_inverse6.svg.png" decoding="async" width="150" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Trigonometric_functions_and_inverse6.svg/225px-Trigonometric_functions_and_inverse6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Trigonometric_functions_and_inverse6.svg/300px-Trigonometric_functions_and_inverse6.svg.png 2x" data-file-width="364" data-file-height="192" /></a> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8e22574702acf2f54e13ae46be44f84e309bad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.226ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(x)}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\operatorname {arccsc}(x))={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\operatorname {arccsc}(x))={\frac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7df66f376887ece366a5170231d4b7516ee21ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.156ex; height:5.176ex;" alt="{\displaystyle \sin(\operatorname {arccsc}(x))={\frac {1}{x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\operatorname {arccsc}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\operatorname {arccsc}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f56f5c4e85f23259e608fd112cca1f1b9b5d8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.792ex; height:7.009ex;" alt="{\displaystyle \cos(\operatorname {arccsc}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\operatorname {arccsc}(x))={\frac {\operatorname {sgn}(x)}{\sqrt {x^{2}-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arccsc}(x))={\frac {\operatorname {sgn}(x)}{\sqrt {x^{2}-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaedfb9df95567bc5adb3aa2ec3a067760c571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.041ex; height:7.009ex;" alt="{\displaystyle \tan(\operatorname {arccsc}(x))={\frac {\operatorname {sgn}(x)}{\sqrt {x^{2}-1}}}}"> </td> <td><a href="/wiki/File:Trigonometric_functions_and_inverse5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Trigonometric_functions_and_inverse5.svg/150px-Trigonometric_functions_and_inverse5.svg.png" decoding="async" width="150" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Trigonometric_functions_and_inverse5.svg/225px-Trigonometric_functions_and_inverse5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Trigonometric_functions_and_inverse5.svg/300px-Trigonometric_functions_and_inverse5.svg.png 2x" data-file-width="297" data-file-height="203" /></a> </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Relationships_among_the_inverse_trigonometric_functions">Relationships among the inverse trigonometric functions</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=10" title="Edit section: Relationships among the inverse trigonometric functions">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arcsine_Arccosine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/168px-Arcsine_Arccosine.svg.png" decoding="async" width="168" height="308" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/252px-Arcsine_Arccosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Arcsine_Arccosine.svg/336px-Arcsine_Arccosine.svg.png 2x" data-file-width="240" data-file-height="440" /></a><figcaption>The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arctangent_Arccotangent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/294px-Arctangent_Arccotangent.svg.png" decoding="async" width="294" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/441px-Arctangent_Arccotangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Arctangent_Arccotangent.svg/588px-Arctangent_Arccotangent.svg.png 2x" data-file-width="420" data-file-height="260" /></a><figcaption>The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Arcsecant_Arccosecant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/294px-Arcsecant_Arccosecant.svg.png" decoding="async" width="294" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/441px-Arcsecant_Arccosecant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Arcsecant_Arccosecant.svg/588px-Arcsecant_Arccosecant.svg.png 2x" data-file-width="420" data-file-height="260" /></a><figcaption>Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.</figcaption></figure> Complementary angles: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)\\[0.5em]\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\\[0.5em]\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)\\[0.5em]\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\\[0.5em]\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec43798232f580abb074cf15f3d77692edd36af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.534ex; margin-bottom: -0.304ex; width:27.82ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)\\[0.5em]\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\\[0.5em]\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)\end{aligned}}}"></dd></dl> Negative arguments: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin(x)\\\operatorname {arccsc}(-x)&=-\operatorname {arccsc}(x)\\\arccos(-x)&=\pi -\arccos(x)\\\operatorname {arcsec}(-x)&=\pi -\operatorname {arcsec}(x)\\\arctan(-x)&=-\arctan(x)\\\operatorname {arccot}(-x)&=\pi -\operatorname {arccot}(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin(x)\\\operatorname {arccsc}(-x)&=-\operatorname {arccsc}(x)\\\arccos(-x)&=\pi -\arccos(x)\\\operatorname {arcsec}(-x)&=\pi -\operatorname {arcsec}(x)\\\arctan(-x)&=-\arctan(x)\\\operatorname {arccot}(-x)&=\pi -\operatorname {arccot}(x)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eae38f47c54d325b9889c2551ed9f42ee887c7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:28.792ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin(x)\\\operatorname {arccsc}(-x)&=-\operatorname {arccsc}(x)\\\arccos(-x)&=\pi -\arccos(x)\\\operatorname {arcsec}(-x)&=\pi -\operatorname {arcsec}(x)\\\arctan(-x)&=-\arctan(x)\\\operatorname {arccot}(-x)&=\pi -\operatorname {arccot}(x)\end{aligned}}}"></dd></dl> Reciprocal arguments: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin \left({\frac {1}{x}}\right)&=\operatorname {arccsc}(x)&\\[0.3em]\operatorname {arccsc} \left({\frac {1}{x}}\right)&=\arcsin(x)&\\[0.3em]\arccos \left({\frac {1}{x}}\right)&=\operatorname {arcsec}(x)&\\[0.3em]\operatorname {arcsec} \left({\frac {1}{x}}\right)&=\arccos(x)&\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x>0\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)-\pi &=-{\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x<0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)&={\frac {\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x>0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)+\pi &={\frac {3\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x<0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.6em 0.6em 0.6em 0.6em 0.6em 0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>π</mi> </mtd> <mtd> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin \left({\frac {1}{x}}\right)&=\operatorname {arccsc}(x)&\\[0.3em]\operatorname {arccsc} \left({\frac {1}{x}}\right)&=\arcsin(x)&\\[0.3em]\arccos \left({\frac {1}{x}}\right)&=\operatorname {arcsec}(x)&\\[0.3em]\operatorname {arcsec} \left({\frac {1}{x}}\right)&=\arccos(x)&\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x>0\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)-\pi &=-{\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x<0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)&={\frac {\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x>0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)+\pi &={\frac {3\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x<0\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2fca7d530f42c84ed92b5895ccbafb013dc6645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.838ex; width:62.732ex; height:54.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin \left({\frac {1}{x}}\right)&=\operatorname {arccsc}(x)&\\[0.3em]\operatorname {arccsc} \left({\frac {1}{x}}\right)&=\arcsin(x)&\\[0.3em]\arccos \left({\frac {1}{x}}\right)&=\operatorname {arcsec}(x)&\\[0.3em]\operatorname {arcsec} \left({\frac {1}{x}}\right)&=\arccos(x)&\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x>0\\[0.3em]\arctan \left({\frac {1}{x}}\right)&=\operatorname {arccot}(x)-\pi &=-{\frac {\pi }{2}}-\arctan(x)\,,{\text{ if }}x<0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)&={\frac {\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x>0\\[0.3em]\operatorname {arccot} \left({\frac {1}{x}}\right)&=\arctan(x)+\pi &={\frac {3\pi }{2}}-\operatorname {arccot}(x)\,,{\text{ if }}x<0\end{aligned}}}"></dd></dl> The identities above can be used with (and derived from) the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399542cf3e9a873371132ffebe579187d48962ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \csc }"> are <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc ={\tfrac {1}{\sin }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>sin</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc ={\tfrac {1}{\sin }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/623a9507aa00d4521c98c92d5f482e1df842271b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.935ex; height:3.509ex;" alt="{\displaystyle \csc ={\tfrac {1}{\sin }}}">), as are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e473a3de151d75296f141f9f482fe59d582a7509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69cf3dc60afb60f19dc5acd070c5b723214192fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.628ex; height:2.009ex;" alt="{\displaystyle \sec ,}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57c91704d9ab0366a6436869e2968491efc5155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024acf2ae8d05b07a2992c33076c71012a0be14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.746ex; height:2.009ex;" alt="{\displaystyle \cot .}"> Useful identities if one only has a fragment of a sine table: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(x)&={\frac {1}{2}}\arccos \left(1-2x^{2}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin(x)&=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)\\\arccos(x)&={\frac {1}{2}}\arccos \left(2x^{2}-1\right)\,,{\text{ if }}0\leq x\leq 1\\\arccos(x)&=\arctan \left({\frac {\sqrt {1-x^{2}}}{x}}\right)\\\arccos(x)&=\arcsin \left({\sqrt {1-x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1{\text{ , from which you get }}\\\arccos &\left({\frac {1-x^{2}}{1+x^{2}}}\right)=\arcsin \left({\frac {2x}{1+x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin &\left({\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\operatorname {sgn}(x)\arcsin(x)\\\arctan(x)&=\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\\\operatorname {arccot}(x)&=\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> , from which you get </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> </mtd> <mtd> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arcsin</mi> </mtd> <mtd> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(x)&={\frac {1}{2}}\arccos \left(1-2x^{2}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin(x)&=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)\\\arccos(x)&={\frac {1}{2}}\arccos \left(2x^{2}-1\right)\,,{\text{ if }}0\leq x\leq 1\\\arccos(x)&=\arctan \left({\frac {\sqrt {1-x^{2}}}{x}}\right)\\\arccos(x)&=\arcsin \left({\sqrt {1-x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1{\text{ , from which you get }}\\\arccos &\left({\frac {1-x^{2}}{1+x^{2}}}\right)=\arcsin \left({\frac {2x}{1+x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin &\left({\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\operatorname {sgn}(x)\arcsin(x)\\\arctan(x)&=\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\\\operatorname {arccot}(x)&=\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0039d679c337a8e095d61c12a4d2b662e75e582d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -28.338ex; width:66.273ex; height:57.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin(x)&={\frac {1}{2}}\arccos \left(1-2x^{2}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin(x)&=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)\\\arccos(x)&={\frac {1}{2}}\arccos \left(2x^{2}-1\right)\,,{\text{ if }}0\leq x\leq 1\\\arccos(x)&=\arctan \left({\frac {\sqrt {1-x^{2}}}{x}}\right)\\\arccos(x)&=\arcsin \left({\sqrt {1-x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1{\text{ , from which you get }}\\\arccos &\left({\frac {1-x^{2}}{1+x^{2}}}\right)=\arcsin \left({\frac {2x}{1+x^{2}}}\right)\,,{\text{ if }}0\leq x\leq 1\\\arcsin &\left({\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\operatorname {sgn}(x)\arcsin(x)\\\arctan(x)&=\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\\\operatorname {arccot}(x)&=\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\end{aligned}}}"></dd></dl> Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real). A useful form that follows directly from the table above is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(x)=\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\,,{\text{ if }}x\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(x)=\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\,,{\text{ if }}x\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b21e76667275aa012e64fdb31a48b4cd74c8c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.681ex; height:7.509ex;" alt="{\displaystyle \arctan(x)=\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\,,{\text{ if }}x\geq 0}">.</dd></dl> It is obtained by recognizing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02dbbd74e6c62a7e56003f54276af9a556b7e264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.507ex; height:7.509ex;" alt="{\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)}">. From the <a href="/wiki/Tangent_half-angle_formula" title="Tangent half-angle formula">half-angle formula</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a9f8dd76ec73197c8301eacc797716c90d7ce5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.079ex; height:4.843ex;" alt="{\displaystyle \tan \left({\tfrac {\theta }{2}}\right)={\tfrac {\sin(\theta )}{1+\cos(\theta )}}}">, we get: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)\\[0.5em]\arccos(x)&=2\arctan \left({\frac {\sqrt {1-x^{2}}}{1+x}}\right)\,,{\text{ if }}-1<x\leq 1\\[0.5em]\arctan(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1+x^{2}}}}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)\\[0.5em]\arccos(x)&=2\arctan \left({\frac {\sqrt {1-x^{2}}}{1+x}}\right)\,,{\text{ if }}-1<x\leq 1\\[0.5em]\arctan(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1+x^{2}}}}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd6a9370a877ca5e198e28b7582bd06b377bdc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:51.332ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)\\[0.5em]\arccos(x)&=2\arctan \left({\frac {\sqrt {1-x^{2}}}{1+x}}\right)\,,{\text{ if }}-1<x\leq 1\\[0.5em]\arctan(x)&=2\arctan \left({\frac {x}{1+{\sqrt {1+x^{2}}}}}\right)\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Arctangent_addition_formula">Arctangent addition formula</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=11" title="Edit section: Arctangent addition formula">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(u)\pm \arctan(v)=\arctan \left({\frac {u\pm v}{1\mp uv}}\right){\pmod {\pi }}\,,\quad uv\neq 1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>±</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mo>±</mo> <mi>v</mi> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>u</mi> <mi>v</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>u</mi> <mi>v</mi> <mo>≠</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(u)\pm \arctan(v)=\arctan \left({\frac {u\pm v}{1\mp uv}}\right){\pmod {\pi }}\,,\quad uv\neq 1\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7b7b6839f595278d3b704ceb358334a7684448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.642ex; height:6.176ex;" alt="{\displaystyle \arctan(u)\pm \arctan(v)=\arctan \left({\frac {u\pm v}{1\mp uv}}\right){\pmod {\pi }}\,,\quad uv\neq 1\,.}"></dd></dl> This is derived from the tangent <a href="/wiki/Angle_sum_and_difference_identities" class="mw-redirect" title="Angle sum and difference identities">addition formula</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\alpha \pm \beta )={\frac {\tan(\alpha )\pm \tan(\beta )}{1\mp \tan(\alpha )\tan(\beta )}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>±</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\alpha \pm \beta )={\frac {\tan(\alpha )\pm \tan(\beta )}{1\mp \tan(\alpha )\tan(\beta )}}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c072c9f9dfe3f5951104b7e6469f39debc2ffcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.345ex; height:6.509ex;" alt="{\displaystyle \tan(\alpha \pm \beta )={\frac {\tan(\alpha )\pm \tan(\beta )}{1\mp \tan(\alpha )\tan(\beta )}}\,,}"></dd></dl> by letting <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\arctan(u)\,,\quad \beta =\arctan(v)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\arctan(u)\,,\quad \beta =\arctan(v)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e30e08f7c0cd50d42a0ea9147b1d8be5aa974bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.802ex; height:2.843ex;" alt="{\displaystyle \alpha =\arctan(u)\,,\quad \beta =\arctan(v)\,.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="In_calculus">In calculus</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=12" title="Edit section: In calculus">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Derivatives_of_inverse_trigonometric_functions">Derivatives of inverse trigonometric functions</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=13" title="Edit section: Derivatives of inverse trigonometric functions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Differentiation_of_trigonometric_functions" title="Differentiation of trigonometric functions">Differentiation of trigonometric functions</a></div> The <a href="/wiki/Derivative" title="Derivative">derivatives</a> for complex values of z are as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin(z)&{}={\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arccos(z)&{}=-{\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arctan(z)&{}={\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arccot}(z)&{}=-{\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcsec}(z)&{}={\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\\{\frac {d}{dz}}\operatorname {arccsc}(z)&{}=-{\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mo>+</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mo>+</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin(z)&{}={\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arccos(z)&{}=-{\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arctan(z)&{}={\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arccot}(z)&{}=-{\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcsec}(z)&{}={\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\\{\frac {d}{dz}}\operatorname {arccsc}(z)&{}=-{\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50aa0820a3e9ce4b0accfce94b992ce0bf850727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -20.338ex; width:49.59ex; height:41.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin(z)&{}={\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arccos(z)&{}=-{\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arctan(z)&{}={\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arccot}(z)&{}=-{\frac {1}{1+z^{2}}}\;;&z&{}\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcsec}(z)&{}={\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\\{\frac {d}{dz}}\operatorname {arccsc}(z)&{}=-{\frac {1}{z^{2}{\sqrt {1-{\frac {1}{z^{2}}}}}}}\;;&z&{}\neq -1,0,+1\end{aligned}}}"></dd></dl> Only for real values of x: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec}(x)&{}={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\\{\frac {d}{dx}}\operatorname {arccsc}(x)&{}=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>;</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec}(x)&{}={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\\{\frac {d}{dx}}\operatorname {arccsc}(x)&{}=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2385485a94b92758573400c6e2bf5c7466647fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:43.645ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec}(x)&{}={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\\{\frac {d}{dx}}\operatorname {arccsc}(x)&{}=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\;;&|x|>1\end{aligned}}}"></dd></dl> These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8e38fd8054b0dbdb14dcc463b5c34d5b97718d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.761ex; height:2.176ex;" alt="{\displaystyle x=\sin \theta }">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/255bad720d1086c6dedcf7d4cb1a5e5fe0a58ad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.157ex; height:3.509ex;" alt="{\textstyle dx/d\theta =\cos \theta ={\sqrt {1-x^{2}}},}"> so <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {d\theta }{dx}}={\frac {1}{dx/d\theta }}={\frac {1}{\sqrt {1-x^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {d\theta }{dx}}={\frac {1}{dx/d\theta }}={\frac {1}{\sqrt {1-x^{2}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6048fdb2209250d458b610885814804d28c8c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.591ex; height:6.676ex;" alt="{\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {d\theta }{dx}}={\frac {1}{dx/d\theta }}={\frac {1}{\sqrt {1-x^{2}}}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Expression_as_definite_integrals">Expression as definite integrals</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=14" title="Edit section: Expression as definite integrals">edit</a>]</div> Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(x)&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arccos(x)&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arctan(x)&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arccot}(x)&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arcsec}(x)&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\pi +\int _{-x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\operatorname {arccsc}(x)&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\int _{-\infty }^{-x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≥</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mtd> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≥</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(x)&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arccos(x)&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arctan(x)&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arccot}(x)&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arcsec}(x)&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\pi +\int _{-x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\operatorname {arccsc}(x)&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\int _{-\infty }^{-x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870fa5404ecb1a2f900cae6a346c11554cc32f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.671ex; width:68.928ex; height:40.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(x)&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arccos(x)&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz\;,&|x|&{}\leq 1\\\arctan(x)&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arccot}(x)&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz\;,\\\operatorname {arcsec}(x)&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\pi +\int _{-x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\operatorname {arccsc}(x)&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz=\int _{-\infty }^{-x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz\;,&x&{}\geq 1\\\end{aligned}}}"></dd></dl> When x equals 1, the integrals with limited domains are <a href="/wiki/Improper_integral" title="Improper integral">improper integrals</a>, but still well-defined. <div class="mw-heading mw-heading3"><h3 id="Infinite_series">Infinite series</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=15" title="Edit section: Infinite series">edit</a>]</div> Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using <a href="/wiki/Power_series" title="Power series">power series</a>, as follows. For arcsine, the series can be derived by expanding its derivative, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f857c726a8e19d36d82ed15e554073973b6e34c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:6.182ex; height:4.509ex;" alt="{\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}}">, as a <a href="/wiki/Binomial_series" title="Binomial series">binomial series</a>, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{1+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{1+z^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f22234ce4e0bd791bb59c74212a792f3940eba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.539ex; height:4.009ex;" alt="{\textstyle {\frac {1}{1+z^{2}}}}"> in a <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a>, and applying the integral definition above (see <a href="/wiki/Leibniz_series" class="mw-redirect" title="Leibniz series">Leibniz series</a>). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(z)&=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n-1)!!}{(2n)!!}}{\frac {z^{2n+1}}{2n+1}}\\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n)!}{(2^{n}n!)^{2}}}{\frac {z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(z)&=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n-1)!!}{(2n)!!}}{\frac {z^{2n+1}}{2n+1}}\\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n)!}{(2^{n}n!)^{2}}}{\frac {z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f778db7f760db059cf12f13ee5c2bf239fbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:62.77ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin(z)&=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n-1)!!}{(2n)!!}}{\frac {z^{2n+1}}{2n+1}}\\[5pt]&=\sum _{n=0}^{\infty }{\frac {(2n)!}{(2^{n}n!)^{2}}}{\frac {z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\end{aligned}}}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\qquad z\neq i,-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mspace width="2em" /> <mi>z</mi> <mo>≠</mo> <mi>i</mi> <mo>,</mo> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\qquad z\neq i,-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ede9759679fb08ed00430503e5d1390495d514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.471ex; height:7.009ex;" alt="{\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\,;\qquad |z|\leq 1\qquad z\neq i,-i}"></dd></dl> Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)=\pi /2-\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)=\pi /2-\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722de2b6a082e123f397a0cc94120a7a5a92ec3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.053ex; height:2.843ex;" alt="{\displaystyle \arccos(x)=\pi /2-\arcsin(x)}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfabcb117471435b5650d56ea95946ec32d7f868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.751ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(x)=\arcsin(1/x)}">, and so on. Another series is given by:<a href="#cite_note-Borwein_2004-14">[13]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left(\arcsin \left({\frac {x}{2}}\right)\right)^{2}=\sum _{n=1}^{\infty }{\frac {x^{2n}}{n^{2}{\binom {2n}{n}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left(\arcsin \left({\frac {x}{2}}\right)\right)^{2}=\sum _{n=1}^{\infty }{\frac {x^{2n}}{n^{2}{\binom {2n}{n}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99ed49c9cfba3e60175afefc4155cc13f6d30a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.606ex; height:7.009ex;" alt="{\displaystyle 2\left(\arcsin \left({\frac {x}{2}}\right)\right)^{2}=\sum _{n=1}^{\infty }{\frac {x^{2n}}{n^{2}{\binom {2n}{n}}}}.}"></dd></dl> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> found a series for the arctangent that converges more quickly than its <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18ea0f644b146b69b19d3029fcb288496064f46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.557ex; height:6.843ex;" alt="{\displaystyle \arctan(z)={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}"><a href="#cite_note-15">[14]</a></dd></dl> (The term in the sum for n = 0 is the <a href="/wiki/Empty_product" title="Empty product">empty product</a>, so is 1.) Alternatively, this can be expressed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}{\frac {z^{2n+1}}{(1+z^{2})^{n+1}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}{\frac {z^{2n+1}}{(1+z^{2})^{n+1}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a291ef0511d04da0a5abb69d2f4bda446c7d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.815ex; height:7.009ex;" alt="{\displaystyle \arctan(z)=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}{\frac {z^{2n+1}}{(1+z^{2})^{n+1}}}.}"></dd></dl> Another series for the arctangent function is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)=i\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\left({\frac {1}{(1+2i/z)^{2n-1}}}-{\frac {1}{(1-2i/z)^{2n-1}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)=i\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\left({\frac {1}{(1+2i/z)^{2n-1}}}-{\frac {1}{(1-2i/z)^{2n-1}}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9717b89ad54a0fbf75440029b99b3e2dae9d8796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.482ex; height:6.843ex;" alt="{\displaystyle \arctan(z)=i\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\left({\frac {1}{(1+2i/z)^{2n-1}}}-{\frac {1}{(1-2i/z)^{2n-1}}}\right),}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i={\sqrt {-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/370c8cebe9634fbfc84c29ea61680b0ad4a1ae0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.807ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}}"> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>.<a href="#cite_note-16">[15]</a> <div class="mw-heading mw-heading4"><h4 id="Continued_fractions_for_arctangent">Continued fractions for arctangent</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=16" title="Edit section: Continued fractions for arctangent">edit</a>]</div> Two alternatives to the power series for arctangent are these <a href="/wiki/Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">generalized continued fractions</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)={\frac {z}{1+{\cfrac {(1z)^{2}}{3-1z^{2}+{\cfrac {(3z)^{2}}{5-3z^{2}+{\cfrac {(5z)^{2}}{7-5z^{2}+{\cfrac {(7z)^{2}}{9-7z^{2}+\ddots }}}}}}}}}}={\frac {z}{1+{\cfrac {(1z)^{2}}{3+{\cfrac {(2z)^{2}}{5+{\cfrac {(3z)^{2}}{7+{\cfrac {(4z)^{2}}{9+\ddots }}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>5</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>−</mo> <mn>5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>7</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mo>−</mo> <mn>7</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)={\frac {z}{1+{\cfrac {(1z)^{2}}{3-1z^{2}+{\cfrac {(3z)^{2}}{5-3z^{2}+{\cfrac {(5z)^{2}}{7-5z^{2}+{\cfrac {(7z)^{2}}{9-7z^{2}+\ddots }}}}}}}}}}={\frac {z}{1+{\cfrac {(1z)^{2}}{3+{\cfrac {(2z)^{2}}{5+{\cfrac {(3z)^{2}}{7+{\cfrac {(4z)^{2}}{9+\ddots }}}}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d4d7246fb4cacd7063161a0c7dab8b4d3e9a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:94.497ex; height:21.676ex;" alt="{\displaystyle \arctan(z)={\frac {z}{1+{\cfrac {(1z)^{2}}{3-1z^{2}+{\cfrac {(3z)^{2}}{5-3z^{2}+{\cfrac {(5z)^{2}}{7-5z^{2}+{\cfrac {(7z)^{2}}{9-7z^{2}+\ddots }}}}}}}}}}={\frac {z}{1+{\cfrac {(1z)^{2}}{3+{\cfrac {(2z)^{2}}{5+{\cfrac {(3z)^{2}}{7+{\cfrac {(4z)^{2}}{9+\ddots }}}}}}}}}}}"></dd></dl> The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2, with each perfect square appearing once. The first was developed by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>; the second by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> utilizing the <a href="/wiki/Gaussian_hypergeometric_series" class="mw-redirect" title="Gaussian hypergeometric series">Gaussian hypergeometric series</a>. <div class="mw-heading mw-heading3"><h3 id="Indefinite_integrals_of_inverse_trigonometric_functions">Indefinite integrals of inverse trigonometric functions</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=17" title="Edit section: Indefinite integrals of inverse trigonometric functions">edit</a>]</div> For real and complex values of z: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \arcsin(z)\,dz&{}=z\,\arcsin(z)+{\sqrt {1-z^{2}}}+C\\\int \arccos(z)\,dz&{}=z\,\arccos(z)-{\sqrt {1-z^{2}}}+C\\\int \arctan(z)\,dz&{}=z\,\arctan(z)-{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arccot}(z)\,dz&{}=z\,\operatorname {arccot}(z)+{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arcsec}(z)\,dz&{}=z\,\operatorname {arcsec}(z)-\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\\\int \operatorname {arccsc}(z)\,dz&{}=z\,\operatorname {arccsc}(z)+\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>z</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>z</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \arcsin(z)\,dz&{}=z\,\arcsin(z)+{\sqrt {1-z^{2}}}+C\\\int \arccos(z)\,dz&{}=z\,\arccos(z)-{\sqrt {1-z^{2}}}+C\\\int \arctan(z)\,dz&{}=z\,\arctan(z)-{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arccot}(z)\,dz&{}=z\,\operatorname {arccot}(z)+{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arcsec}(z)\,dz&{}=z\,\operatorname {arcsec}(z)-\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\\\int \operatorname {arccsc}(z)\,dz&{}=z\,\operatorname {arccsc}(z)+\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e2dde92bb82231c4326e45ce8b50e7298688bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.005ex; width:59.896ex; height:39.176ex;" alt="{\displaystyle {\begin{aligned}\int \arcsin(z)\,dz&{}=z\,\arcsin(z)+{\sqrt {1-z^{2}}}+C\\\int \arccos(z)\,dz&{}=z\,\arccos(z)-{\sqrt {1-z^{2}}}+C\\\int \arctan(z)\,dz&{}=z\,\arctan(z)-{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arccot}(z)\,dz&{}=z\,\operatorname {arccot}(z)+{\frac {1}{2}}\ln \left(1+z^{2}\right)+C\\\int \operatorname {arcsec}(z)\,dz&{}=z\,\operatorname {arcsec}(z)-\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\\\int \operatorname {arccsc}(z)\,dz&{}=z\,\operatorname {arccsc}(z)+\ln \left[z\left(1+{\sqrt {\frac {z^{2}-1}{z^{2}}}}\right)\right]+C\end{aligned}}}"></dd></dl> For real x ≥ 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55463426b688a92d18ff0198dd8a56bf7c1e80b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:54.963ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\end{aligned}}}"></dd></dl> For all real x not between -1 and 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9ae0956619a4a16f085dd7f44e3de491e81b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:60.378ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C\end{aligned}}}"></dd></dl> The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the <a href="#Derivatives">derivatives</a> of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the <a href="/wiki/Inverse_hyperbolic_functions#Logarithmic_representation" title="Inverse hyperbolic functions">inverse hyperbolic functions</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {arcosh} (|x|)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {arcosh} (|x|)+C\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>arcosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>arcosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {arcosh} (|x|)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {arcosh} (|x|)+C\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e94aa603d89955db702862fd88ec6e49d9bcc316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:48.278ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\int \operatorname {arcsec}(x)\,dx&{}=x\,\operatorname {arcsec}(x)-\operatorname {arcosh} (|x|)+C\\\int \operatorname {arccsc}(x)\,dx&{}=x\,\operatorname {arccsc}(x)+\operatorname {arcosh} (|x|)+C\\\end{aligned}}}"></dd></dl> The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above. All of these antiderivatives can be derived using <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a> and the simple derivative forms shown above. <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=18" title="Edit section: Example">edit</a>]</div> Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int u\,dv=uv-\int v\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>u</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mi>u</mi> <mi>v</mi> <mo>−</mo> <mo>∫</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int u\,dv=uv-\int v\,du}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddb2871f48408d699da1d94af2076a15008989a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.678ex; height:5.676ex;" alt="{\displaystyle \int u\,dv=uv-\int v\,du}"> (i.e. <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>), set <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}u&=\arcsin(x)&dv&=dx\\du&={\frac {dx}{\sqrt {1-x^{2}}}}&v&=x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>d</mi> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi>v</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u&=\arcsin(x)&dv&=dx\\du&={\frac {dx}{\sqrt {1-x^{2}}}}&v&=x\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d423169c683f702bad48de97fbb408477bb73eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.069ex; margin-bottom: -0.269ex; width:28.575ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}u&=\arcsin(x)&dv&=dx\\du&={\frac {dx}{\sqrt {1-x^{2}}}}&v&=x\end{aligned}}}"></dd></dl> Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)-\int {\frac {x}{\sqrt {1-x^{2}}}}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)-\int {\frac {x}{\sqrt {1-x^{2}}}}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33506e7ed1a37ad25d1890958f2b183c15dd23e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.078ex; height:6.509ex;" alt="{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)-\int {\frac {x}{\sqrt {1-x^{2}}}}\,dx,}"></dd></dl> which by the simple <a href="/wiki/Integration_by_substitution" title="Integration by substitution">substitution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=1-x^{2},\ dw=-2x\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mi>w</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w=1-x^{2},\ dw=-2x\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d235b18fbc90305e4a470bdd60db037ad6e484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.975ex; height:3.009ex;" alt="{\displaystyle w=1-x^{2},\ dw=-2x\,dx}"> yields the final result: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aca6ab38a9c44c197ca561b42f8584c1715d70a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.688ex; height:5.676ex;" alt="{\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Extension_to_the_complex_plane">Extension to the complex plane</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=19" title="Edit section: Extension to the complex plane">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Riemann_surface_for_Arg_of_ArcTan_of_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Riemann_surface_for_Arg_of_ArcTan_of_x.svg/220px-Riemann_surface_for_Arg_of_ArcTan_of_x.svg.png" decoding="async" width="220" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Riemann_surface_for_Arg_of_ArcTan_of_x.svg/330px-Riemann_surface_for_Arg_of_ArcTan_of_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Riemann_surface_for_Arg_of_ArcTan_of_x.svg/440px-Riemann_surface_for_Arg_of_ArcTan_of_x.svg.png 2x" data-file-width="440" data-file-height="421" /></a><figcaption>A <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a> for the argument of the relation tan z = x. The orange sheet in the middle is the principal sheet representing arctan x. The blue sheet above and green sheet below are displaced by 2π and −2π respectively.</figcaption></figure> Since the inverse trigonometric functions are <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and <a href="/wiki/Branch_point" title="Branch point">branch points</a>. One possible way of defining the extension is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)=\int _{0}^{z}{\frac {dx}{1+x^{2}}}\quad z\neq -i,+i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mo>+</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)=\int _{0}^{z}{\frac {dx}{1+x^{2}}}\quad z\neq -i,+i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b0459d0246b804242b583e80dc925bc2062ba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.302ex; height:5.843ex;" alt="{\displaystyle \arctan(z)=\int _{0}^{z}{\frac {dx}{1+x^{2}}}\quad z\neq -i,+i}"></dd></dl> where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the <a href="/wiki/Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut. The arcsine function may then be defined as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(z)=\arctan \left({\frac {z}{\sqrt {1-z^{2}}}}\right)\quad z\neq -1,+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(z)=\arctan \left({\frac {z}{\sqrt {1-z^{2}}}}\right)\quad z\neq -1,+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074e0d5ee56596723519bbaa503a86f0b2e35f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.897ex; height:7.509ex;" alt="{\displaystyle \arcsin(z)=\arctan \left({\frac {z}{\sqrt {1-z^{2}}}}\right)\quad z\neq -1,+1}"></dd></dl> where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(z)={\frac {\pi }{2}}-\arcsin(z)\quad z\neq -1,+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(z)={\frac {\pi }{2}}-\arcsin(z)\quad z\neq -1,+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3adc1e8748bd1f9ba6051b6b3330eedce6430554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.566ex; height:4.676ex;" alt="{\displaystyle \arccos(z)={\frac {\pi }{2}}-\arcsin(z)\quad z\neq -1,+1}"></dd></dl> which has the same cut as arcsin; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(z)={\frac {\pi }{2}}-\arctan(z)\quad z\neq -i,i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mi>i</mi> <mo>,</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(z)={\frac {\pi }{2}}-\arctan(z)\quad z\neq -i,i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a874845a0365d2c8141e908462c9ed4c777503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.53ex; height:4.676ex;" alt="{\displaystyle \operatorname {arccot}(z)={\frac {\pi }{2}}-\arctan(z)\quad z\neq -i,i}"></dd></dl> which has the same cut as arctan; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(z)=\arccos \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(z)=\arccos \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bb204b4af0113f33d21f7c051a40a7e3fca621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.401ex; height:6.176ex;" alt="{\displaystyle \operatorname {arcsec}(z)=\arccos \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}"></dd></dl> where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(z)=\arcsin \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="1em" /> <mi>z</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(z)=\arcsin \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2946069afdcc33032436c472143b30df0c19c434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.146ex; height:6.176ex;" alt="{\displaystyle \operatorname {arccsc}(z)=\arcsin \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1}"></dd></dl> which has the same cut as arcsec. <div class="mw-heading mw-heading3"><h3 id="Logarithmic_forms">Logarithmic forms</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=20" title="Edit section: Logarithmic forms">edit</a>]</div> These functions may also be expressed using <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithms</a>. This extends their <a href="/wiki/Domain_of_a_function" title="Domain of a function">domains</a> to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(z)&{}=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)=i\ln \left({\sqrt {1-z^{2}}}-iz\right)&{}=\operatorname {arccsc} \left({\frac {1}{z}}\right)\\[10pt]\arccos(z)&{}=-i\ln \left(i{\sqrt {1-z^{2}}}+z\right)={\frac {\pi }{2}}-\arcsin(z)&{}=\operatorname {arcsec} \left({\frac {1}{z}}\right)\\[10pt]\arctan(z)&{}=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)=-{\frac {i}{2}}\ln \left({\frac {1+iz}{1-iz}}\right)&{}=\operatorname {arccot} \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccot}(z)&{}=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)=-{\frac {i}{2}}\ln \left({\frac {iz-1}{iz+1}}\right)&{}=\arctan \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arcsec}(z)&{}=-i\ln \left(i{\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {1}{z}}\right)={\frac {\pi }{2}}-\operatorname {arccsc}(z)&{}=\arccos \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccsc}(z)&{}=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)=i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}-{\frac {i}{z}}\right)&{}=\arcsin \left({\frac {1}{z}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 1.3em 1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccsc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arcsec</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>i</mi> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(z)&{}=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)=i\ln \left({\sqrt {1-z^{2}}}-iz\right)&{}=\operatorname {arccsc} \left({\frac {1}{z}}\right)\\[10pt]\arccos(z)&{}=-i\ln \left(i{\sqrt {1-z^{2}}}+z\right)={\frac {\pi }{2}}-\arcsin(z)&{}=\operatorname {arcsec} \left({\frac {1}{z}}\right)\\[10pt]\arctan(z)&{}=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)=-{\frac {i}{2}}\ln \left({\frac {1+iz}{1-iz}}\right)&{}=\operatorname {arccot} \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccot}(z)&{}=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)=-{\frac {i}{2}}\ln \left({\frac {iz-1}{iz+1}}\right)&{}=\arctan \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arcsec}(z)&{}=-i\ln \left(i{\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {1}{z}}\right)={\frac {\pi }{2}}-\operatorname {arccsc}(z)&{}=\arccos \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccsc}(z)&{}=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)=i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}-{\frac {i}{z}}\right)&{}=\arcsin \left({\frac {1}{z}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b511be341b52fcd0c0660f3a4b1e5a164bfcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.171ex; width:78.994ex; height:49.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(z)&{}=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)=i\ln \left({\sqrt {1-z^{2}}}-iz\right)&{}=\operatorname {arccsc} \left({\frac {1}{z}}\right)\\[10pt]\arccos(z)&{}=-i\ln \left(i{\sqrt {1-z^{2}}}+z\right)={\frac {\pi }{2}}-\arcsin(z)&{}=\operatorname {arcsec} \left({\frac {1}{z}}\right)\\[10pt]\arctan(z)&{}=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)=-{\frac {i}{2}}\ln \left({\frac {1+iz}{1-iz}}\right)&{}=\operatorname {arccot} \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccot}(z)&{}=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)=-{\frac {i}{2}}\ln \left({\frac {iz-1}{iz+1}}\right)&{}=\arctan \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arcsec}(z)&{}=-i\ln \left(i{\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {1}{z}}\right)={\frac {\pi }{2}}-\operatorname {arccsc}(z)&{}=\arccos \left({\frac {1}{z}}\right)\\[10pt]\operatorname {arccsc}(z)&{}=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)=i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}-{\frac {i}{z}}\right)&{}=\arcsin \left({\frac {1}{z}}\right)\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Generalization">Generalization</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=21" title="Edit section: Generalization">edit</a>]</div> Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> to form a right triangle in the complex plane. Algebraically, this gives us: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce^{i\theta }=c\cos(\theta )+ic\sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>c</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ce^{i\theta }=c\cos(\theta )+ic\sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b96b1cd319048f9536db8c2979bf2b2d81268c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.957ex; height:3.176ex;" alt="{\displaystyle ce^{i\theta }=c\cos(\theta )+ic\sin(\theta )}"></dd></dl> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce^{i\theta }=a+ib}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ce^{i\theta }=a+ib}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f445125bfe8fd5884c9d5604c398433f405b10be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.63ex; height:2.843ex;" alt="{\displaystyle ce^{i\theta }=a+ib}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is the adjacent side, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> is the opposite side, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> is the hypotenuse. From here, we can solve for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e^{\ln(c)+i\theta }&=a+ib\\\ln c+i\theta &=\ln(a+ib)\\\theta &=\operatorname {Im} \left(\ln(a+ib)\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mi>c</mi> <mo>+</mo> <mi>i</mi> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{\ln(c)+i\theta }&=a+ib\\\ln c+i\theta &=\ln(a+ib)\\\theta &=\operatorname {Im} \left(\ln(a+ib)\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20378ffade5e9dbbb81cc0fc93cb0c950d79e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.12ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}e^{\ln(c)+i\theta }&=a+ib\\\ln c+i\theta &=\ln(a+ib)\\\theta &=\operatorname {Im} \left(\ln(a+ib)\right)\end{aligned}}}"></dd></dl> or <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-i\ln \left({\frac {a+ib}{c}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-i\ln \left({\frac {a+ib}{c}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72383d1816287229c19192ff9ff49a206285c057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.253ex; height:6.176ex;" alt="{\displaystyle \theta =-i\ln \left({\frac {a+ib}{c}}\right)}"></dd></dl> Simply taking the imaginary part works for any real-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, but if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(a+bi)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(a+bi)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a19206c3079c5f23675b705849eff0dd6ea5af5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.619ex; height:2.843ex;" alt="{\displaystyle \ln(a+bi)}"> also removes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the <a href="/wiki/Pythagorean_Theorem" class="mw-redirect" title="Pythagorean Theorem">Pythagorean Theorem</a> relation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}=c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}"></dd></dl> The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> that result from plugging the values into the equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mrow> <mi>c</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f95b2bf8af7e2dbfe7e909eb82ced96991c155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.159ex; height:4.843ex;" alt="{\displaystyle \theta =-i\ln \left({\tfrac {a+ib}{c}}\right)}"> above and simplifying. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&a&&b&&c&&-i\ln \left({\frac {a+ib}{c}}\right)&&\theta &&\theta _{a,b\in \mathbb {R} }\\\arcsin(z)\ \ &{\sqrt {1-z^{2}}}&&z&&1&&-i\ln \left({\frac {{\sqrt {1-z^{2}}}+iz}{1}}\right)&&=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-z^{2}}}+iz\right)\right)\\\arccos(z)\ \ &z&&{\sqrt {1-z^{2}}}&&1&&-i\ln \left({\frac {z+i{\sqrt {1-z^{2}}}}{1}}\right)&&=-i\ln \left(z+{\sqrt {z^{2}-1}}\right)&&\operatorname {Im} \left(\ln \left(z+{\sqrt {z^{2}-1}}\right)\right)\\\arctan(z)\ \ &1&&z&&{\sqrt {1+z^{2}}}&&-i\ln \left({\frac {1+iz}{\sqrt {1+z^{2}}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)&&\operatorname {Im} \left(\ln \left(1+iz\right)\right)\\\operatorname {arccot}(z)\ \ &z&&1&&{\sqrt {z^{2}+1}}&&-i\ln \left({\frac {z+i}{\sqrt {z^{2}+1}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)&&\operatorname {Im} \left(\ln \left(z+i\right)\right)\\\operatorname {arcsec}(z)\ \ &1&&{\sqrt {z^{2}-1}}&&z&&-i\ln \left({\frac {1+i{\sqrt {z^{2}-1}}}{z}}\right)&&=-i\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)&&\operatorname {Im} \left(\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)\right)\\\operatorname {arccsc}(z)\ \ &{\sqrt {z^{2}-1}}&&1&&z&&-i\ln \left({\frac {{\sqrt {z^{2}-1}}+i}{z}}\right)&&=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)\right)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>a</mi> </mtd> <mtd /> <mtd> <mi>b</mi> </mtd> <mtd /> <mtd> <mi>c</mi> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>θ</mi> </mtd> <mtd /> <mtd> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo>−</mo> <mi>z</mi> </mrow> <mrow> <mi>i</mi> <mo>+</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> </mrow> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>+</mo> <mi>i</mi> </mrow> <mi>z</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&a&&b&&c&&-i\ln \left({\frac {a+ib}{c}}\right)&&\theta &&\theta _{a,b\in \mathbb {R} }\\\arcsin(z)\ \ &{\sqrt {1-z^{2}}}&&z&&1&&-i\ln \left({\frac {{\sqrt {1-z^{2}}}+iz}{1}}\right)&&=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-z^{2}}}+iz\right)\right)\\\arccos(z)\ \ &z&&{\sqrt {1-z^{2}}}&&1&&-i\ln \left({\frac {z+i{\sqrt {1-z^{2}}}}{1}}\right)&&=-i\ln \left(z+{\sqrt {z^{2}-1}}\right)&&\operatorname {Im} \left(\ln \left(z+{\sqrt {z^{2}-1}}\right)\right)\\\arctan(z)\ \ &1&&z&&{\sqrt {1+z^{2}}}&&-i\ln \left({\frac {1+iz}{\sqrt {1+z^{2}}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)&&\operatorname {Im} \left(\ln \left(1+iz\right)\right)\\\operatorname {arccot}(z)\ \ &z&&1&&{\sqrt {z^{2}+1}}&&-i\ln \left({\frac {z+i}{\sqrt {z^{2}+1}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)&&\operatorname {Im} \left(\ln \left(z+i\right)\right)\\\operatorname {arcsec}(z)\ \ &1&&{\sqrt {z^{2}-1}}&&z&&-i\ln \left({\frac {1+i{\sqrt {z^{2}-1}}}{z}}\right)&&=-i\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)&&\operatorname {Im} \left(\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)\right)\\\operatorname {arccsc}(z)\ \ &{\sqrt {z^{2}-1}}&&1&&z&&-i\ln \left({\frac {{\sqrt {z^{2}-1}}+i}{z}}\right)&&=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)\right)\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4d3ce30fa9a7387579409cecaaff1123f1b9a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -25.671ex; width:134.985ex; height:52.509ex;" alt="{\displaystyle {\begin{aligned}&a&&b&&c&&-i\ln \left({\frac {a+ib}{c}}\right)&&\theta &&\theta _{a,b\in \mathbb {R} }\\\arcsin(z)\ \ &{\sqrt {1-z^{2}}}&&z&&1&&-i\ln \left({\frac {{\sqrt {1-z^{2}}}+iz}{1}}\right)&&=-i\ln \left({\sqrt {1-z^{2}}}+iz\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-z^{2}}}+iz\right)\right)\\\arccos(z)\ \ &z&&{\sqrt {1-z^{2}}}&&1&&-i\ln \left({\frac {z+i{\sqrt {1-z^{2}}}}{1}}\right)&&=-i\ln \left(z+{\sqrt {z^{2}-1}}\right)&&\operatorname {Im} \left(\ln \left(z+{\sqrt {z^{2}-1}}\right)\right)\\\arctan(z)\ \ &1&&z&&{\sqrt {1+z^{2}}}&&-i\ln \left({\frac {1+iz}{\sqrt {1+z^{2}}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {i-z}{i+z}}\right)&&\operatorname {Im} \left(\ln \left(1+iz\right)\right)\\\operatorname {arccot}(z)\ \ &z&&1&&{\sqrt {z^{2}+1}}&&-i\ln \left({\frac {z+i}{\sqrt {z^{2}+1}}}\right)&&=-{\frac {i}{2}}\ln \left({\frac {z+i}{z-i}}\right)&&\operatorname {Im} \left(\ln \left(z+i\right)\right)\\\operatorname {arcsec}(z)\ \ &1&&{\sqrt {z^{2}-1}}&&z&&-i\ln \left({\frac {1+i{\sqrt {z^{2}-1}}}{z}}\right)&&=-i\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)&&\operatorname {Im} \left(\ln \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}-1}}\right)\right)\\\operatorname {arccsc}(z)\ \ &{\sqrt {z^{2}-1}}&&1&&z&&-i\ln \left({\frac {{\sqrt {z^{2}-1}}+i}{z}}\right)&&=-i\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)&&\operatorname {Im} \left(\ln \left({\sqrt {1-{\frac {1}{z^{2}}}}}+{\frac {i}{z}}\right)\right)\\\end{aligned}}}"></dd></dl> The particular form of the simplified expression can cause the output to differ from the <a href="#Principal_values">usual principal branch</a> of each of the inverse trig functions. The formulations given will output the usual principal branch when using the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781cef7f2317c18794eeaaddeef4073aadd51b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.898ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} \left(\ln z\right)\in (-\pi ,\pi ]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8544bd56a919a5a8390d94487ac7821f74019a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.837ex; height:3.009ex;" alt="{\displaystyle \operatorname {Re} \left({\sqrt {z}}\right)\geq 0}"> principal branch for every function except arccotangent in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> column. Arccotangent in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> column will output on its usual principal branch by using the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec91a56edab8fb5d96781734a583e4139620c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.082ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} \left(\ln z\right)\in [0,2\pi )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64acf6ee879aa8f6c70eaac2d28249facc0282cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.87ex; height:3.009ex;" alt="{\displaystyle \operatorname {Im} \left({\sqrt {z}}\right)\geq 0}"> convention. In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">, the definitions allow for <a href="/wiki/Hyperbolic_angle" title="Hyperbolic angle">hyperbolic angles</a> as outputs and can be used to further define the <a href="/wiki/Inverse_hyperbolic_functions" title="Inverse hyperbolic functions">inverse hyperbolic functions</a>. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function. <div class="mw-heading mw-heading4"><h4 id="Example_proof">Example proof</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=22" title="Edit section: Example proof">edit</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(\phi )&=z\\\phi &=\arcsin(z)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\phi )&=z\\\phi &=\arcsin(z)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6536d655b6311b315e32e6ed5416318289f8d76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.76ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sin(\phi )&=z\\\phi &=\arcsin(z)\end{aligned}}}"></dd></dl> Using the <a href="/wiki/Trigonometric_functions#Relationship_to_exponential_function_(Euler's_formula)" title="Trigonometric functions">exponential definition of sine</a>, and letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =e^{i\phi },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi =e^{i\phi },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5994c092d2b672378568b886a2f79585dc1093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.638ex; height:3.009ex;" alt="{\displaystyle \xi =e^{i\phi },}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}z&={\frac {e^{i\phi }-e^{-i\phi }}{2i}}\\[10mu]2iz&=\xi -{\frac {1}{\xi }}\\[5mu]0&=\xi ^{2}-2iz\xi -1\\[5mu]\xi &=iz\pm {\sqrt {1-z^{2}}}\\[5mu]\phi &=-i\ln \left(iz\pm {\sqrt {1-z^{2}}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.856em 0.578em 0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>i</mi> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ξ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ξ</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>z</mi> <mi>ξ</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>ξ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>i</mi> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ϕ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z&={\frac {e^{i\phi }-e^{-i\phi }}{2i}}\\[10mu]2iz&=\xi -{\frac {1}{\xi }}\\[5mu]0&=\xi ^{2}-2iz\xi -1\\[5mu]\xi &=iz\pm {\sqrt {1-z^{2}}}\\[5mu]\phi &=-i\ln \left(iz\pm {\sqrt {1-z^{2}}}\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed03e8cf773fa44cc78823d65f7d82f41276fb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:27.818ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}z&={\frac {e^{i\phi }-e^{-i\phi }}{2i}}\\[10mu]2iz&=\xi -{\frac {1}{\xi }}\\[5mu]0&=\xi ^{2}-2iz\xi -1\\[5mu]\xi &=iz\pm {\sqrt {1-z^{2}}}\\[5mu]\phi &=-i\ln \left(iz\pm {\sqrt {1-z^{2}}}\right)\end{aligned}}}"></dd></dl> (the positive branch is chosen) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =\arcsin(z)=-i\ln \left(iz+{\sqrt {1-z^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\arcsin(z)=-i\ln \left(iz+{\sqrt {1-z^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1d620af6b384b58787d500f1d5ad0a2c3d55b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.357ex; height:4.843ex;" alt="{\displaystyle \phi =\arcsin(z)=-i\ln \left(iz+{\sqrt {1-z^{2}}}\right)}"></dd></dl> <table style="text-align:center;"> <caption><a href="/wiki/Domain_coloring" title="Domain coloring">Color wheel graphs</a> of inverse trigonometric functions in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> </caption> <tbody><tr> <td><a href="/wiki/File:Complex_Arcsine.svg" class="mw-file-description" title="Arcsine of z in the complex plane."><img alt="Arcsine of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Complex_Arcsine.svg/275px-Complex_Arcsine.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Complex_Arcsine.svg/413px-Complex_Arcsine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Complex_Arcsine.svg/550px-Complex_Arcsine.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td> <td><a href="/wiki/File:Complex_Arccosine.svg" class="mw-file-description" title="Arccosine of z in the complex plane."><img alt="Arccosine of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Complex_Arccosine.svg/275px-Complex_Arccosine.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Complex_Arccosine.svg/413px-Complex_Arccosine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Complex_Arccosine.svg/550px-Complex_Arccosine.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td> <td><a href="/wiki/File:Complex_Arctangent.svg" class="mw-file-description" title="Arctangent of z in the complex plane."><img alt="Arctangent of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Complex_Arctangent.svg/275px-Complex_Arctangent.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Complex_Arctangent.svg/413px-Complex_Arctangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Complex_Arctangent.svg/550px-Complex_Arctangent.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee6682ca9597bbe4c2a4288c5cc07a85f6dc9f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.86ex; height:2.843ex;" alt="{\displaystyle \arcsin(z)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc22b22fad84588b35c004301571a76763b69fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.115ex; height:2.843ex;" alt="{\displaystyle \arccos(z)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5639d7225fe861c9dc7b9d17420d53b474cbe9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.364ex; height:2.843ex;" alt="{\displaystyle \arctan(z)}"> </td></tr></tbody></table> <table style="text-align:center;"> <caption> </caption> <tbody><tr> <td><a href="/wiki/File:Complex_Arccosecant.svg" class="mw-file-description" title="Arccosecant of z in the complex plane."><img alt="Arccosecant of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Complex_Arccosecant.svg/275px-Complex_Arccosecant.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Complex_Arccosecant.svg/413px-Complex_Arccosecant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Complex_Arccosecant.svg/550px-Complex_Arccosecant.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td> <td><a href="/wiki/File:Complex_Arcsecant.svg" class="mw-file-description" title="Arcsecant of z in the complex plane."><img alt="Arcsecant of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Complex_Arcsecant.svg/275px-Complex_Arcsecant.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Complex_Arcsecant.svg/413px-Complex_Arcsecant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Complex_Arcsecant.svg/550px-Complex_Arcsecant.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td> <td><a href="/wiki/File:Complex_Arccotangent.svg" class="mw-file-description" title="Arccotangent of z in the complex plane."><img alt="Arccotangent of z in the complex plane." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_Arccotangent.svg/275px-Complex_Arccotangent.svg.png" decoding="async" width="275" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_Arccotangent.svg/413px-Complex_Arccotangent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Complex_Arccotangent.svg/550px-Complex_Arccotangent.svg.png 2x" data-file-width="540" data-file-height="360" /></a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccsc}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccsc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccsc}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03e7565a043ebf7a7ddfa814cfea67edf0ab28f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccsc}(z)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcsec}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcsec}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82a7fce4bcb85464efa221ca744a6352c412eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.985ex; height:2.843ex;" alt="{\displaystyle \operatorname {arcsec}(z)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arccot}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arccot}(z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33429aa4786f03503d41646358f0d7dc63d9aed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.103ex; height:2.843ex;" alt="{\displaystyle \operatorname {arccot}(z)}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=23" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Finding_the_angle_of_a_right_triangle">Finding the angle of a right triangle</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=24" title="Edit section: Finding the angle of a right triangle">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Trigonometry_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/220px-Trigonometry_triangle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/330px-Trigonometry_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/440px-Trigonometry_triangle.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>A <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> with sides relative to an angle at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> point.</figcaption></figure> Inverse trigonometric functions are useful when trying to determine the remaining two angles of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>opposite</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>adjacent</mtext> <mtext>hypotenuse</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67aab6a7270ad083bf6cbed9957568a63fffdb70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.52ex; height:6.176ex;" alt="{\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text{hypotenuse}}}\right).}"></dd></dl> Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the <a href="/wiki/Pythagorean_Theorem" class="mw-redirect" title="Pythagorean Theorem">Pythagorean Theorem</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}=h^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=h^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffbeeaf7a7dd047e6cb3fbc7b2e2f0f1aa8c098b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.668ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=h^{2}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>opposite</mtext> <mtext>adjacent</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9446611506baae7f1d99e1cc9e3fdc16c3e8e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.538ex; height:6.176ex;" alt="{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)\,.}"></dd></dl> For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)=\arctan \left({\frac {\text{rise}}{\text{run}}}\right)=\arctan \left({\frac {8}{20}}\right)\approx 21.8^{\circ }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>opposite</mtext> <mtext>adjacent</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>rise</mtext> <mtext>run</mtext> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>20</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <msup> <mn>21.8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)=\arctan \left({\frac {\text{rise}}{\text{run}}}\right)=\arctan \left({\frac {8}{20}}\right)\approx 21.8^{\circ }\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079e4b1403005011a65e7c9389bbc05a9b338056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.3ex; height:6.176ex;" alt="{\displaystyle \theta =\arctan \left({\frac {\text{opposite}}{\text{adjacent}}}\right)=\arctan \left({\frac {\text{rise}}{\text{run}}}\right)=\arctan \left({\frac {8}{20}}\right)\approx 21.8^{\circ }\,.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="In_computer_science_and_engineering">In computer science and engineering</h3>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=25" title="Edit section: In computer science and engineering">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Two-argument_variant_of_arctangent">Two-argument variant of arctangent</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=26" title="Edit section: Two-argument variant of arctangent">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Atan2" title="Atan2">atan2</a></div> The two-argument <a href="/wiki/Atan2" title="Atan2">atan2</a> function computes the arctangent of y / x given y and x, but with a range of (−π, π]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering. In terms of the standard arctan function, that is with range of (−<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠π/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠π/2⁠), it can be expressed as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>π</mi> </mtd> <mtd> <mspace width="1em" /> <mi>y</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>π</mi> </mtd> <mtd> <mspace width="1em" /> <mi>y</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>y</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>y</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>undefined</mtext> </mrow> </mtd> <mtd> <mspace width="1em" /> <mi>y</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ff56e5d97eb006f879f8e0d04f6629c9b8cb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.956ex; margin-bottom: -0.215ex; width:48.805ex; height:19.509ex;" alt="{\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&\quad x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &\quad y\geq 0,\;x<0\\\arctan \left({\frac {y}{x}}\right)-\pi &\quad y<0,\;x<0\\{\frac {\pi }{2}}&\quad y>0,\;x=0\\-{\frac {\pi }{2}}&\quad y<0,\;x=0\\{\text{undefined}}&\quad y=0,\;x=0\end{cases}}}"></dd></dl> It also equals the <a href="/wiki/Principal_value" title="Principal value">principal value</a> of the <a href="/wiki/Arg_(mathematics)" class="mw-redirect" title="Arg (mathematics)">argument</a> of the <a href="/wiki/Complex_number" title="Complex number">complex number</a> x + iy. This limited version of the function above may also be defined using the <a href="/wiki/Tangent_half-angle_formula" title="Tangent half-angle formula">tangent half-angle formulae</a> as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93e84a5cca1a410cff6a1f871c4965b1750b66bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.577ex; height:7.509ex;" alt="{\displaystyle \operatorname {atan2} (y,x)=2\arctan \left({\frac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)}"></dd></dl> provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use. The above argument order (y, x) seems to be the most common, and in particular is used in <a href="/wiki/ISO_standard" class="mw-redirect" title="ISO standard">ISO standards</a> such as the <a href="/wiki/C_(programming_language)" title="C (programming language)">C programming language</a>, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at <a href="/wiki/Atan2#Realizations_of_the_function_in_common_computer_languages" title="Atan2">atan2</a>. <div class="mw-heading mw-heading4"><h4 id="Arctangent_function_with_location_parameter">Arctangent function with location parameter</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=27" title="Edit section: Arctangent function with location parameter">edit</a>]</div> In many applications<a href="#cite_note-17">[16]</a> the solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> of the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\tan(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\tan(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6aba9dc9e6af08b9365cc58202cabac122b6f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.753ex; height:2.843ex;" alt="{\displaystyle x=\tan(y)}"> is to come as close as possible to a given value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty <\eta <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>η</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty <\eta <\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00619db6caaf8362025e328d275f31153510a03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.822ex; height:2.509ex;" alt="{\displaystyle -\infty <\eta <\infty }">. The adequate solution is produced by the parameter modified arctangent function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arctan _{\eta }(x):=\arctan(x)+\pi \,\operatorname {rni} \left({\frac {\eta -\arctan(x)}{\pi }}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>arctan</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mspace width="thinmathspace" /> <mi>rni</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>η</mi> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arctan _{\eta }(x):=\arctan(x)+\pi \,\operatorname {rni} \left({\frac {\eta -\arctan(x)}{\pi }}\right)\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0217e1c4ae1b91db78a7d0bb126b899a123f82f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.972ex; height:6.343ex;" alt="{\displaystyle y=\arctan _{\eta }(x):=\arctan(x)+\pi \,\operatorname {rni} \left({\frac {\eta -\arctan(x)}{\pi }}\right)\,.}"></dd></dl> The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rni} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rni</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rni} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11af24c3265b955efdc5d8a90d041a0e32fe36e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.851ex; height:2.176ex;" alt="{\displaystyle \operatorname {rni} }"> rounds to the nearest integer. <div class="mw-heading mw-heading4"><h4 id="Numerical_accuracy">Numerical accuracy</h4>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=28" title="Edit section: Numerical accuracy">edit</a>]</div> For angles near 0 and π, arccosine is <a href="/wiki/Ill-conditioned" class="mw-redirect" title="Ill-conditioned">ill-conditioned</a>, and similarly with arcsine for angles near −π/2 and π/2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods.<a href="#cite_note-Gade_2010-18">[17]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=29" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine distribution</a></li> <li><a href="/wiki/Inverse_exsecant" class="mw-redirect" title="Inverse exsecant">Inverse exsecant</a></li> <li><a href="/wiki/Inverse_versine" class="mw-redirect" title="Inverse versine">Inverse versine</a></li> <li><a href="/wiki/Inverse_hyperbolic_functions" title="Inverse hyperbolic functions">Inverse hyperbolic functions</a></li> <li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">List of integrals of inverse trigonometric functions</a></li> <li><a href="/wiki/List_of_trigonometric_identities" title="List of trigonometric identities">List of trigonometric identities</a></li> <li><a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">Trigonometric function</a></li> <li><a href="/wiki/Trigonometric_functions_of_matrices" title="Trigonometric functions of matrices">Trigonometric functions of matrices</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=30" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><a href="#cite_ref-12">^</a> The expression "LHS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> RHS" would not have been written. To clarify, suppose that it is written "LHS <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> RHS" where LHS (which abbreviates left hand side) and RHS are both statements that can individually be either be true or false. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> are some given and fixed numbers and if the following is written: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =s\,\iff \,\theta =\arctan(s)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>k</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =s\,\iff \,\theta =\arctan(s)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31f4b4c431b23f3f4bb8f2e29a78085faf5ca853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.136ex; height:2.843ex;" alt="{\displaystyle \tan \theta =s\,\iff \,\theta =\arctan(s)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }"> then LHS is the statement "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f078184d98dd7239fc2cb3ca550367c2d43bb9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.026ex; height:2.176ex;" alt="{\displaystyle \tan \theta =s}">". Depending on what specific values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> have, this LHS statement can either be true or false. For instance, LHS is true if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7903b8069a44c70f6f96511675bdd9a4ff200ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}"> (because in this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =\tan 0=s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =\tan 0=s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc53b9738d9d58a28fedb073c5e0a92fa35e131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.034ex; height:2.176ex;" alt="{\displaystyle \tan \theta =\tan 0=s}">) but LHS is false if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2127cfe51e8f7f982b180284ca0912f37ff38356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=2}"> (because in this case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =\tan 0=s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \theta =\tan 0=s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc53b9738d9d58a28fedb073c5e0a92fa35e131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.034ex; height:2.176ex;" alt="{\displaystyle \tan \theta =\tan 0=s}"> which is not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2127cfe51e8f7f982b180284ca0912f37ff38356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=2}">); more generally, LHS is false if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325558648e6f31f7d1050afead1d4d289f3936d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.998ex; height:2.676ex;" alt="{\displaystyle s\neq 0.}"> Similarly, RHS is the statement "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan(s)+\pi k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>π</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan(s)+\pi k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4208733509f57afef97fa9ec05886b07df843db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.938ex; height:2.843ex;" alt="{\displaystyle \theta =\arctan(s)+\pi k}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }">". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> have). The logical equality symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> means that (a) if the LHS statement is true then the RHS statement is also necessarily true, and moreover (b) if the LHS statement is false then the RHS statement is also necessarily false. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\iff \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\iff \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9403ae29dc51b393f3c77ce83444c7183d582b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.381ex; height:1.843ex;" alt="{\displaystyle \,\iff \,}"> also means that (c) if the RHS statement is true then the LHS statement is also necessarily true, and moreover (d) if the RHS statement is false then the LHS statement is also necessarily false. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=31" title="Edit section: References">edit</a>]</div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAbramowitzStegun1972" class="citation book cs1"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene A.</a>, eds. (1972). <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofmathe000abra">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1972&rft.isbn=978-0-486-61272-0&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhandbookofmathe000abra&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Hall_1909-1">^ <a href="#cite_ref-Hall_1909_1-0">a</a> <a href="#cite_ref-Hall_1909_1-1">b</a> <a href="#cite_ref-Hall_1909_1-2">c</a> <a href="#cite_ref-Hall_1909_1-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallFrink1909" class="citation book cs1 cs1-prop-location-test cs1-prop-long-vol">Hall, Arthur Graham; Frink, Fred Goodrich (Jan 1909). <a rel="nofollow" class="external text" href="https://archive.org/stream/planetrigonometr00hallrich#page/n30/mode/1up">"Chapter II. The Acute Angle [14] Inverse trigonometric functions"</a>. Written at Ann Arbor, Michigan, USA. Trigonometry. Vol. Part I: Plane Trigonometry. New York, USA: <a href="/wiki/Henry_Holt_and_Company" title="Henry Holt and Company">Henry Holt and Company</a> / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. <q>[…] <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style>α = arcsin m: It is frequently read "<a href="/wiki/Arc-sine" class="mw-redirect" title="Arc-sine">arc-sine</a> m" or "<a href="/wiki/Anti-sine" class="mw-redirect" title="Anti-sine">anti-sine</a> m," since two mutually inverse functions are said each to be the <a href="/wiki/Anti-function" class="mw-redirect" title="Anti-function">anti-function</a> of the other. […] A similar symbolic relation holds for the other <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">α = sin-1m, is still found in English and American texts. The notation <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">α = inv sin m is perhaps better still on account of its general applicability. […]</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+II.+The+Acute+Angle+%5B14%5D+Inverse+trigonometric+functions&rft.btitle=Trigonometry&rft.place=New+York%2C+USA&rft.pages=15&rft.pub=Henry+Holt+and+Company+%2F+Norwood+Press+%2F+J.+S.+Cushing+Co.+-+Berwick+%26+Smith+Co.%2C+Norwood%2C+Massachusetts%2C+USA&rft.date=1909-01&rft.aulast=Hall&rft.aufirst=Arthur+Graham&rft.au=Frink%2C+Fred+Goodrich&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fplanetrigonometr00hallrich%23page%2Fn30%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-cyclometric-2"><a href="#cite_ref-cyclometric_2-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1924" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, Felix</a> (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). Vol. 1 (3rd ed.). Berlin: J. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementarmathematik+vom+h%C3%B6heren+Standpunkt+aus%3A+Arithmetik%2C+Algebra%2C+Analysis&rft.place=Berlin&rft.edition=3rd&rft.pub=J.+Springer&rft.date=1924&rft.aulast=Klein&rft.aufirst=Felix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> Translated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1932" class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=8KuoxgykfbkC">Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis</a>. Translated by Hedrick, E. R.; Noble, C. A. Macmillan. 1932. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43480-3" title="Special:BookSources/978-0-486-43480-3"><bdi>978-0-486-43480-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Mathematics+from+an+Advanced+Standpoint%3A+Arithmetic%2C+Algebra%2C+Analysis&rft.pub=Macmillan&rft.date=1932&rft.isbn=978-0-486-43480-3&rft.aulast=Klein&rft.aufirst=Felix&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8KuoxgykfbkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-arcus-3"><a href="#cite_ref-arcus_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkel1994" class="citation book cs1"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a> (1994) [1987]. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopaedia of Mathematics</a> (unabridged reprint ed.). <a href="/wiki/Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a> / <a href="/wiki/Springer_Science_%26_Business_Media" class="mw-redirect" title="Springer Science & Business Media">Springer Science & Business Media</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-155608010-4" title="Special:BookSources/978-155608010-4"><bdi>978-155608010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+of+Mathematics&rft.edition=unabridged+reprint&rft.pub=Kluwer+Academic+Publishers+%2F+Springer+Science+%26+Business+Media&rft.date=1994&rft.isbn=978-155608010-4&rft.aulast=Hazewinkel&rft.aufirst=Michiel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> <div class="paragraphbreak" style="margin-top:0.5em"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBronshteinSemendyayevMusiolMühlig" class="citation book cs1">Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner. "Cyclometric or Inverse Trigonometric Functions". Handbook of Mathematics (6th ed.). Berlin: Springer. § 2.8, pp. 85–89. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-663-46221-8">10.1007/978-3-663-46221-8</a> (inactive 1 Nov 2024).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cyclometric+or+Inverse+Trigonometric+Functions&rft.btitle=Handbook+of+Mathematics&rft.place=Berlin&rft.pages=%3Cspan+class%3D%22nowrap%22%3E%C2%A7+2.8%3C%2Fspan%3E%2C+pp.-85-89&rft.edition=6th&rft.pub=Springer&rft_id=info%3Adoi%2F10.1007%2F978-3-663-46221-8&rft.aulast=Bronshtein&rft.aufirst=I.+N.&rft.au=Semendyayev%2C+K.+A.&rft.au=Musiol%2C+Gerhard&rft.au=M%C3%BChlig%2C+Heiner&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>) <div class="paragraphbreak" style="margin-top:0.5em"></div> However, the term "arcus function" can also refer to the function giving the <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of a complex number, sometimes called the arcus. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/InverseTrigonometricFunctions.html">"Inverse Trigonometric Functions"</a>. mathworld.wolfram.com. Retrieved 2020-08-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Inverse+Trigonometric+Functions&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FInverseTrigonometricFunctions.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-Americana_1912-5"><a href="#cite_ref-Americana_1912_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeachRines1912" class="citation book cs1">Beach, Frederick Converse; Rines, George Edwin, eds. (1912). "Inverse trigonometric functions". <a href="/wiki/The_Americana" class="mw-redirect" title="The Americana">The Americana: a universal reference library</a>. Vol. 21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Inverse+trigonometric+functions&rft.btitle=The+Americana%3A+a+universal+reference+library&rft.date=1912&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCook,_John_D.2021" class="citation web cs1">Cook, John D. (11 Feb 2021). <a rel="nofollow" class="external text" href="https://www.johndcook.com/blog/2021/02/11/trig-across-languages">"Trig functions across programming languages"</a>. johndcook.com (blog). Retrieved 2021-03-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=johndcook.com&rft.atitle=Trig+functions+across+programming+languages&rft.date=2021-02-11&rft.au=Cook%2C+John+D.&rft_id=https%3A%2F%2Fwww.johndcook.com%2Fblog%2F2021%2F02%2F11%2Ftrig-across-languages&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-Cajori-7"><a href="#cite_ref-Cajori_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1919" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1919). <a rel="nofollow" class="external text" href="https://archive.org/details/ahistorymathema02cajogoog">A History of Mathematics</a> (2 ed.). New York, NY: <a href="/wiki/The_Macmillan_Company" class="mw-redirect" title="The Macmillan Company">The Macmillan Company</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/ahistorymathema02cajogoog/page/n284">272</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=New+York%2C+NY&rft.pages=272&rft.edition=2&rft.pub=The+Macmillan+Company&rft.date=1919&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fahistorymathema02cajogoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-Herschel_1813-8"><a href="#cite_ref-Herschel_1813_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerschel1813" class="citation journal cs1"><a href="/wiki/John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1813). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qpRJAAAAYAAJ&pg=PA8">"On a remarkable Application of Cotes's Theorem"</a>. 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Royal Society, London: 8. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">10.1098/rstl.1813.0005</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions&rft.atitle=On+a+remarkable+Application+of+Cotes%27s+Theorem&rft.volume=103&rft.issue=1&rft.pages=8&rft.date=1813&rft_id=info%3Adoi%2F10.1098%2Frstl.1813.0005&rft.aulast=Herschel&rft.aufirst=John+Frederick+William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqpRJAAAAYAAJ%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://brilliant.org/wiki/inverse-trigonometric-functions/">"Inverse trigonometric functions"</a>. 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Mineola, New York, USA: <a href="/wiki/Dover_Publications,_Inc." class="mw-redirect" title="Dover Publications, Inc.">Dover Publications, Inc.</a> p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalhand00korn_849/page/n828">811</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-41147-7" title="Special:BookSources/978-0-486-41147-7"><bdi>978-0-486-41147-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=21.2.-4.+Inverse+Trigonometric+Functions&rft.btitle=Mathematical+handbook+for+scientists+and+engineers%3A+Definitions%2C+theorems%2C+and+formulars+for+reference+and+review&rft.place=Mineola%2C+New+York%2C+USA&rft.pages=811&rft.edition=3&rft.pub=Dover+Publications%2C+Inc.&rft.date=2000&rft.isbn=978-0-486-41147-7&rft.aulast=Korn&rft.aufirst=Grandino+Arthur&rft.au=Korn%2C+Theresa+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalhand00korn_849&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-Bhatti_1999-11"><a href="#cite_ref-Bhatti_1999_11-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhattiNawab-ud-DinAhmedYousuf1999" class="citation book cs1">Bhatti, Sanaullah; Nawab-ud-Din; Ahmed, Bashir; Yousuf, S. 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Peters</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/experimentationm00borw_656/page/n60">51</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-136-9" title="Special:BookSources/978-1-56881-136-9"><bdi>978-1-56881-136-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Experimentation+in+Mathematics%3A+Computational+Paths+to+Discovery&rft.place=Wellesley%2C+MA%2C+USA&rft.pages=51&rft.edition=1&rft.pub=A.+K.+Peters&rft.date=2004&rft.isbn=978-1-56881-136-9&rft.aulast=Borwein&rft.aufirst=Jonathan&rft.au=Bailey%2C+David&rft.au=Gingersohn%2C+Roland&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fexperimentationm00borw_656&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHwang_Chien-Lih2005" class="citation cs2">Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette, 89 (516): 469–470, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0025557200178404">10.1017/S0025557200178404</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123395287">123395287</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=An+elementary+derivation+of+Euler%27s+series+for+the+arctangent+function&rft.volume=89&rft.issue=516&rft.pages=469-470&rft.date=2005&rft_id=info%3Adoi%2F10.1017%2FS0025557200178404&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123395287%23id-name%3DS2CID&rft.au=Hwang+Chien-Lih&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS._M._Abrarov_and_B._M._Quine2018" class="citation cs2">S. M. Abrarov and B. M. Quine (2018), "A formula for pi involving nested radicals", The Ramanujan Journal, 46 (3): 657–665, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.07713">1610.07713</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11139-018-9996-8">10.1007/s11139-018-9996-8</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119150623">119150623</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Ramanujan+Journal&rft.atitle=A+formula+for+pi+involving+nested+radicals&rft.volume=46&rft.issue=3&rft.pages=657-665&rft.date=2018&rft_id=info%3Aarxiv%2F1610.07713&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119150623%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11139-018-9996-8&rft.au=S.+M.+Abrarov+and+B.+M.+Quine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> when a time varying angle crossing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed24d6d7c48b32f1d8ce709a3bf2717d9b0a59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.465ex; height:2.843ex;" alt="{\displaystyle \pm \pi /2}"> should be mapped by a smooth line instead of a saw toothed one (robotics, astromomy, angular movement in general)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </li> <li id="cite_note-Gade_2010-18"><a href="#cite_ref-Gade_2010_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGade2010" class="citation journal cs1">Gade, Kenneth (2010). <a rel="nofollow" class="external text" href="http://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf">"A non-singular horizontal position representation"</a> (PDF). The Journal of Navigation. 63 (3). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>: 395–417. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010JNav...63..395G">2010JNav...63..395G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0373463309990415">10.1017/S0373463309990415</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Navigation&rft.atitle=A+non-singular+horizontal+position+representation&rft.volume=63&rft.issue=3&rft.pages=395-417&rft.date=2010&rft_id=info%3Adoi%2F10.1017%2FS0373463309990415&rft_id=info%3Abibcode%2F2010JNav...63..395G&rft.aulast=Gade&rft.aufirst=Kenneth&rft_id=http%3A%2F%2Fwww.navlab.net%2FPublications%2FA_Nonsingular_Horizontal_Position_Representation.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=32" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/InverseTangent.html">"Inverse Tangent"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Inverse+Tangent&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FInverseTangent.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInverse+trigonometric+functions" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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Inverse trigonometric functions - Wikipedia