Τριγωνομετρία - Βικιπαίδεια

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Γενικά"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Γενικά</span> </div> </a> <button aria-controls="toc-Γενικά-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Γενικά υποενότητας</span> </button> <ul id="toc-Γενικά-sublist" class="vector-toc-list"> <li id="toc-Η_επέκταση_των_ορισμών" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Η_επέκταση_των_ορισμών"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Η επέκταση των ορισμών</span> </div> </a> <ul id="toc-Η_επέκταση_των_ορισμών-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Υπολογισμός_τριγωνομετρικών_συναρτήσεων" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Υπολογισμός_τριγωνομετρικών_συναρτήσεων"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Υπολογισμός τριγωνομετρικών συναρτήσεων</span> </div> </a> <ul id="toc-Υπολογισμός_τριγωνομετρικών_συναρτήσεων-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ταυτότητες" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ταυτότητες"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ταυτότητες</span> </div> </a> <button aria-controls="toc-Ταυτότητες-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Εναλλαγή Ταυτότητες υποενότητας</span> </button> <ul id="toc-Ταυτότητες-sublist" class="vector-toc-list"> <li id="toc-Σχετικές_με_το_Πυθαγόρειο_θεώρημα" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Σχετικές_με_το_Πυθαγόρειο_θεώρημα"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Σχετικές με το Πυθαγόρειο θεώρημα</span> </div> </a> <ul id="toc-Σχετικές_με_το_Πυθαγόρειο_θεώρημα-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Βασικές" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Βασικές"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Βασικές</span> </div> </a> <ul id="toc-Βασικές-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Αθροίσματος_και_διαφοράς_γωνιών" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Αθροίσματος_και_διαφοράς_γωνιών"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Αθροίσματος και διαφοράς γωνιών</span> </div> </a> <ul id="toc-Αθροίσματος_και_διαφοράς_γωνιών-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Πολλαπλασιασμού_γωνιών" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Πολλαπλασιασμού_γωνιών"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Πολλαπλασιασμού γωνιών</span> </div> </a> <ul id="toc-Πολλαπλασιασμού_γωνιών-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Αθροίσματος_και_διαφοράς_τριγωνομετρικών_συναρτήσεων" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Αθροίσματος_και_διαφοράς_τριγωνομετρικών_συναρτήσεων"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Αθροίσματος και διαφοράς τριγωνομετρικών συναρτήσεων</span> </div> </a> <ul id="toc-Αθροίσματος_και_διαφοράς_τριγωνομετρικών_συναρτήσεων-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Σε_ένα_τρίγωνο" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Σε_ένα_τρίγωνο"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Σε ένα τρίγωνο</span> </div> </a> <ul id="toc-Σε_ένα_τρίγωνο-sublist" class="vector-toc-list"> <li id="toc-Νόμος_των_ημιτόνων" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Νόμος_των_ημιτόνων"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.1</span> <span>Νόμος των ημιτόνων</span> </div> </a> <ul id="toc-Νόμος_των_ημιτόνων-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Νόμος_των_συνημιτόνων" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Νόμος_των_συνημιτόνων"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.2</span> <span>Νόμος των συνημιτόνων</span> </div> </a> <ul id="toc-Νόμος_των_συνημιτόνων-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Νόμος_των_εφαπτομένων" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Νόμος_των_εφαπτομένων"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.3</span> <span>Νόμος των εφαπτομένων</span> </div> </a> <ul id="toc-Νόμος_των_εφαπτομένων-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Το_θεώρημα_των_προβολών" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Το_θεώρημα_των_προβολών"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Το θεώρημα των προβολών</span> </div> </a> <ul id="toc-Το_θεώρημα_των_προβολών-sublist" class="vector-toc-list"> <li id="toc-Τύπος_του_Όιλερ" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Τύπος_του_Όιλερ"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7.1</span> <span>Τύπος του Όιλερ</span> </div> </a> <ul id="toc-Τύπος_του_Όιλερ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Διάφορες_σχέσεις" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Διάφορες_σχέσεις"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Διάφορες σχέσεις</span> </div> </a> <ul id="toc-Διάφορες_σχέσεις-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Σφαιρική_τριγωνομετρία" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Σφαιρική_τριγωνομετρία"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Σφαιρική τριγωνομετρία</span> </div> </a> <ul id="toc-Σφαιρική_τριγωνομετρία-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Εφαρμογές_της_τριγωνομετρίας" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Εφαρμογές_της_τριγωνομετρίας"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Εφαρμογές της τριγωνομετρίας</span> </div> </a> <ul id="toc-Εφαρμογές_της_τριγωνομετρίας-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Δείτε_επίσης" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Δείτε_επίσης"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Δείτε επίσης</span> </div> </a> <ul id="toc-Δείτε_επίσης-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Παραπομπές" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Παραπομπές"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Παραπομπές</span> </div> </a> <ul id="toc-Παραπομπές-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Εξωτερικοί_σύνδεσμοι" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Εξωτερικοί_σύνδεσμοι"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Εξωτερικοί σύνδεσμοι</span> </div> </a> <ul id="toc-Εξωτερικοί_σύνδεσμοι-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="Περιεχόμενα" 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="Εναλλαγή του πίνακα περιεχομένων" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Εναλλαγή του πίνακα περιεχομένων</span> </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"><span class="mw-page-title-main">Τριγωνομετρία</span></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="Μεταβείτε σε ένα λήμμα σε άλλη γλώσσα. Διαθέσιμο σε 139 γλώσσες" > <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-139" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">139 γλώσσες</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Driehoeksmeting" title="Driehoeksmeting – Αφρικάανς" lang="af" hreflang="af" data-title="Driehoeksmeting" data-language-autonym="Afrikaans" data-language-local-name="Αφρικάανς" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Γερμανικά Ελβετίας" lang="gsw" hreflang="gsw" data-title="Trigonometrie" data-language-autonym="Alemannisch" data-language-local-name="Γερμανικά Ελβετίας" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B5%E1%88%AA%E1%8C%8E%E1%8A%96%E1%88%9C%E1%89%B5%E1%88%AA" title="ትሪጎኖሜትሪ – Αμχαρικά" lang="am" hreflang="am" data-title="ትሪጎኖሜትሪ" data-language-autonym="አማርኛ" data-language-local-name="Αμχαρικά" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría – Αραγονικά" lang="an" hreflang="an" data-title="Trigonometría" data-language-autonym="Aragonés" data-language-local-name="Αραγονικά" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%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%A4%BF" title="त्रिकोणमिति – Ανγκικά" lang="anp" hreflang="anp" data-title="त्रिकोणमिति" data-language-autonym="अंगिका" data-language-local-name="Ανγκικά" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="حساب المثلثات – Αραβικά" lang="ar" hreflang="ar" data-title="حساب المثلثات" data-language-autonym="العربية" data-language-local-name="Αραβικά" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="حساب المثلثات – Egyptian Arabic" lang="arz" hreflang="arz" data-title="حساب المثلثات" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A7%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" title="ত্ৰিকোণমিতি – Ασαμικά" lang="as" hreflang="as" data-title="ত্ৰিকোণমিতি" data-language-autonym="অসমীয়া" data-language-local-name="Ασαμικά" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría – Αστουριανά" lang="ast" hreflang="ast" data-title="Trigonometría" data-language-autonym="Asturianu" data-language-local-name="Αστουριανά" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Triqonometriya" title="Triqonometriya – Αζερμπαϊτζανικά" lang="az" hreflang="az" data-title="Triqonometriya" data-language-autonym="Azərbaycanca" data-language-local-name="Αζερμπαϊτζανικά" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%88%DA%86%E2%80%8C%D8%A8%D9%88%D8%AC%D8%A7%D9%82_%D8%A8%DB%8C%D9%84%DB%8C%D9%85%DB%8C" title="اوچ‌بوجاق بیلیمی – South Azerbaijani" lang="azb" hreflang="azb" data-title="اوچ‌بوجاق بیلیمی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.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%D1%8F" title="Тригонометрия – Μπασκίρ" lang="ba" hreflang="ba" data-title="Тригонометрия" data-language-autonym="Башҡортса" data-language-local-name="Μπασκίρ" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Tr%C4%97guonuometr%C4%97j%C4%97" title="Trėguonuometrėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Trėguonuometrėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya – Central Bikol" lang="bcl" hreflang="bcl" data-title="Trigonometriya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Трыганаметрыя – Λευκορωσικά" lang="be" hreflang="be" data-title="Трыганаметрыя" data-language-autonym="Беларуская" data-language-local-name="Λευκορωσικά" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Трыганамэтрыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Трыганамэтрыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.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%D1%8F" title="Тригонометрия – Βουλγαρικά" lang="bg" hreflang="bg" data-title="Тригонометрия" data-language-autonym="Български" data-language-local-name="Βουλγαρικά" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%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" title="ত্রিকোণমিতি – Βεγγαλικά" lang="bn" hreflang="bn" data-title="ত্রিকোণমিতি" data-language-autonym="বাংলা" data-language-local-name="Βεγγαλικά" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Trigonometriezh" title="Trigonometriezh – Βρετονικά" lang="br" hreflang="br" data-title="Trigonometriezh" data-language-autonym="Brezhoneg" data-language-local-name="Βρετονικά" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Βοσνιακά" lang="bs" hreflang="bs" data-title="Trigonometrija" data-language-autonym="Bosanski" data-language-local-name="Βοσνιακά" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Καταλανικά" lang="ca" hreflang="ca" data-title="Trigonometria" data-language-autonym="Català" data-language-local-name="Καταλανικά" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%D8%B2%D8%A7%D9%86%DB%8C" title="سێگۆشەزانی – Κεντρικά Κουρδικά" lang="ckb" hreflang="ckb" data-title="سێگۆشەزانی" data-language-autonym="کوردی" data-language-local-name="Κεντρικά Κουρδικά" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Trigunumitria" title="Trigunumitria – Κορσικανικά" lang="co" hreflang="co" data-title="Trigunumitria" data-language-autonym="Corsu" data-language-local-name="Κορσικανικά" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Τσεχικά" lang="cs" hreflang="cs" data-title="Trigonometrie" data-language-autonym="Čeština" data-language-local-name="Τσεχικά" class="interlanguage-link-target"><span>Čeština</span></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" title="Тригонометри – Τσουβασικά" lang="cv" hreflang="cv" data-title="Тригонометри" data-language-autonym="Чӑвашла" data-language-local-name="Τσουβασικά" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Trigonometreg" title="Trigonometreg – Ουαλικά" lang="cy" hreflang="cy" data-title="Trigonometreg" data-language-autonym="Cymraeg" data-language-local-name="Ουαλικά" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Δανικά" lang="da" hreflang="da" data-title="Trigonometri" data-language-autonym="Dansk" data-language-local-name="Δανικά" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Γερμανικά" lang="de" hreflang="de" data-title="Trigonometrie" data-language-autonym="Deutsch" data-language-local-name="Γερμανικά" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Trigonometr%C3%AE" title="Trigonometrî – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Trigonometrî" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Trigonometry" title="Trigonometry – Αγγλικά" lang="en" hreflang="en" data-title="Trigonometry" data-language-autonym="English" data-language-local-name="Αγγλικά" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Trigonometrio" title="Trigonometrio – Εσπεράντο" lang="eo" hreflang="eo" data-title="Trigonometrio" data-language-autonym="Esperanto" data-language-local-name="Εσπεράντο" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría – Ισπανικά" lang="es" hreflang="es" data-title="Trigonometría" data-language-autonym="Español" data-language-local-name="Ισπανικά" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Trigonomeetria" title="Trigonomeetria – Εσθονικά" lang="et" hreflang="et" data-title="Trigonomeetria" data-language-autonym="Eesti" data-language-local-name="Εσθονικά" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Βασκικά" lang="eu" hreflang="eu" data-title="Trigonometria" data-language-autonym="Euskara" data-language-local-name="Βασκικά" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Extremaduran" lang="ext" hreflang="ext" data-title="Trigonometria" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA" title="مثلثات – Περσικά" lang="fa" hreflang="fa" data-title="مثلثات" data-language-autonym="فارسی" data-language-local-name="Περσικά" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Φινλανδικά" lang="fi" hreflang="fi" data-title="Trigonometria" data-language-autonym="Suomi" data-language-local-name="Φινλανδικά" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Trigonomeetri%C3%A4" title="Trigonomeetriä – Võro" lang="vro" hreflang="vro" data-title="Trigonomeetriä" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Φεροϊκά" lang="fo" hreflang="fo" data-title="Trigonometri" data-language-autonym="Føroyskt" data-language-local-name="Φεροϊκά" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Trigonom%C3%A9trie" title="Trigonométrie – Γαλλικά" lang="fr" hreflang="fr" data-title="Trigonométrie" data-language-autonym="Français" data-language-local-name="Γαλλικά" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Trigonometrii" title="Trigonometrii – Βόρεια Φριζιανά" lang="frr" hreflang="frr" data-title="Trigonometrii" data-language-autonym="Nordfriisk" data-language-local-name="Βόρεια Φριζιανά" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Triant%C3%A1nacht" title="Triantánacht – Ιρλανδικά" lang="ga" hreflang="ga" data-title="Triantánacht" data-language-autonym="Gaeilge" data-language-local-name="Ιρλανδικά" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%B8" title="三角學 – Gan" lang="gan" hreflang="gan" data-title="三角學" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Trigonom%C3%A9tri" title="Trigonométri – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Trigonométri" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Trigonometr%C3%ADa" title="Trigonometría – Γαλικιανά" lang="gl" hreflang="gl" data-title="Trigonometría" data-language-autonym="Galego" data-language-local-name="Γαλικιανά" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%BF%E0%AA%95%E0%AB%8B%E0%AA%A3%E0%AA%AE%E0%AA%BF%E0%AA%A4%E0%AA%BF" title="ત્રિકોણમિતિ – Γκουτζαρατικά" lang="gu" hreflang="gu" data-title="ત્રિકોણમિતિ" data-language-autonym="ગુજરાતી" data-language-local-name="Γκουτζαρατικά" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="טריגונומטריה – Εβραϊκά" lang="he" hreflang="he" data-title="טריגונומטריה" data-language-autonym="עברית" data-language-local-name="Εβραϊκά" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%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%A4%BF" title="त्रिकोणमिति – Χίντι" lang="hi" hreflang="hi" data-title="त्रिकोणमिति" data-language-autonym="हिन्दी" data-language-local-name="Χίντι" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Trigonometry" title="Trigonometry – Fiji Hindi" lang="hif" hreflang="hif" data-title="Trigonometry" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Κροατικά" lang="hr" hreflang="hr" data-title="Trigonometrija" data-language-autonym="Hrvatski" data-language-local-name="Κροατικά" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Ουγγρικά" lang="hu" hreflang="hu" data-title="Trigonometria" data-language-autonym="Magyar" data-language-local-name="Ουγγρικά" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B5%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%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Եռանկյունաչափություն – Αρμενικά" lang="hy" hreflang="hy" data-title="Եռանկյունաչափություն" data-language-autonym="Հայերեն" data-language-local-name="Αρμενικά" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Ιντερλίνγκουα" lang="ia" hreflang="ia" data-title="Trigonometria" data-language-autonym="Interlingua" data-language-local-name="Ιντερλίνγκουα" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Ιμπάν" lang="iba" hreflang="iba" data-title="Trigonometri" data-language-autonym="Jaku Iban" data-language-local-name="Ιμπάν" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Ινδονησιακά" lang="id" hreflang="id" data-title="Trigonometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="Ινδονησιακά" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Ιλόκο" lang="ilo" hreflang="ilo" data-title="Trigonometria" data-language-autonym="Ilokano" data-language-local-name="Ιλόκο" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Trigonometrio" title="Trigonometrio – Ίντο" lang="io" hreflang="io" data-title="Trigonometrio" data-language-autonym="Ido" data-language-local-name="Ίντο" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hornafr%C3%A6%C3%B0i" title="Hornafræði – Ισλανδικά" lang="is" hreflang="is" data-title="Hornafræði" data-language-autonym="Íslenska" data-language-local-name="Ισλανδικά" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Ιταλικά" lang="it" hreflang="it" data-title="Trigonometria" data-language-autonym="Italiano" data-language-local-name="Ιταλικά" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E6%B3%95" title="三角法 – Ιαπωνικά" lang="ja" hreflang="ja" data-title="三角法" data-language-autonym="日本語" data-language-local-name="Ιαπωνικά" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Chriganamichri" title="Chriganamichri – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Chriganamichri" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Trigonom%C3%A8tri" title="Trigonomètri – Ιαβανικά" lang="jv" hreflang="jv" data-title="Trigonomètri" data-language-autonym="Jawa" data-language-local-name="Ιαβανικά" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%98%E1%83%92%E1%83%9D%E1%83%9C%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90" title="ტრიგონომეტრია – Γεωργιανά" lang="ka" hreflang="ka" data-title="ტრიგონომეტრია" data-language-autonym="ქართული" data-language-local-name="Γεωργιανά" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya – Κάρα-Καλπάκ" lang="kaa" hreflang="kaa" data-title="Trigonometriya" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Κάρα-Καλπάκ" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/K%C9%94l%C9%94m%C9%A9%C5%8B_naadozo_t%CA%8A_pilinzi_maz%CA%8A%CA%8A" title="Kɔlɔmɩŋ naadozo tʊ pilinzi mazʊʊ – Kabiye" lang="kbp" hreflang="kbp" data-title="Kɔlɔmɩŋ naadozo tʊ pilinzi mazʊʊ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.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%D1%8F" title="Тригонометрия – Καζακικά" lang="kk" hreflang="kk" data-title="Тригонометрия" data-language-autonym="Қазақша" data-language-local-name="Καζακικά" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%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" title="ត្រីកោណមាត្រ – Χμερ" lang="km" hreflang="km" data-title="ត្រីកោណមាត្រ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Χμερ" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81%EB%B2%95" title="삼각법 – Κορεατικά" lang="ko" hreflang="ko" data-title="삼각법" data-language-autonym="한국어" data-language-local-name="Κορεατικά" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/S%C3%AAgo%C5%9Fenas%C3%AE" title="Sêgoşenasî – Κουρδικά" lang="ku" hreflang="ku" data-title="Sêgoşenasî" data-language-autonym="Kurdî" data-language-local-name="Κουρδικά" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.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%D1%8F" title="Тригонометрия – Κιργιζικά" lang="ky" hreflang="ky" data-title="Тригонометрия" data-language-autonym="Кыргызча" data-language-local-name="Κιργιζικά" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Λατινικά" lang="la" hreflang="la" data-title="Trigonometria" data-language-autonym="Latina" data-language-local-name="Λατινικά" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Trigonometria" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Essomampuyisatu_(Trigonometry)" title="Essomampuyisatu (Trigonometry) – Γκάντα" lang="lg" hreflang="lg" data-title="Essomampuyisatu (Trigonometry)" data-language-autonym="Luganda" data-language-local-name="Γκάντα" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Goniometrie" title="Goniometrie – Λιμβουργιανά" lang="li" hreflang="li" data-title="Goniometrie" data-language-autonym="Limburgs" data-language-local-name="Λιμβουργιανά" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Lombard" lang="lmo" hreflang="lmo" data-title="Trigonometria" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%84%E0%BA%95%E0%BA%A1%E0%BA%B8%E0%BA%A1" title="ໄຕມຸມ – Λαοτινά" lang="lo" hreflang="lo" data-title="ໄຕມຸມ" data-language-autonym="ລາວ" data-language-local-name="Λαοτινά" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Λιθουανικά" lang="lt" hreflang="lt" data-title="Trigonometrija" data-language-autonym="Lietuvių" data-language-local-name="Λιθουανικά" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Λετονικά" lang="lv" hreflang="lv" data-title="Trigonometrija" data-language-autonym="Latviešu" data-language-local-name="Λετονικά" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.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%D1%98%D0%B0" title="Тригонометрија – Σλαβομακεδονικά" lang="mk" hreflang="mk" data-title="Тригонометрија" data-language-autonym="Македонски" data-language-local-name="Σλαβομακεδονικά" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B4%A3%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF" title="ത്രികോണമിതി – Μαλαγιαλαμικά" lang="ml" hreflang="ml" data-title="ത്രികോണമിതി" data-language-autonym="മലയാളം" data-language-local-name="Μαλαγιαλαμικά" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%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" title="त्रिकोणमिती – Μαραθικά" lang="mr" hreflang="mr" data-title="त्रिकोणमिती" data-language-autonym="मराठी" data-language-local-name="Μαραθικά" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Μαλαισιανά" lang="ms" hreflang="ms" data-title="Trigonometri" data-language-autonym="Bahasa Melayu" data-language-local-name="Μαλαισιανά" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%BC%E1%80%AE%E1%80%82%E1%80%AD%E1%80%AF%E1%80%94%E1%80%AD%E1%80%AF%E1%80%99%E1%80%B1%E1%80%90%E1%80%BC%E1%80%AE" title="တြီဂိုနိုမေတြီ – Βιρμανικά" lang="my" hreflang="my" data-title="တြီဂိုနိုမေတြီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Βιρμανικά" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Κάτω Γερμανικά" lang="nds" hreflang="nds" data-title="Trigonometrie" data-language-autonym="Plattdüütsch" data-language-local-name="Κάτω Γερμανικά" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%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%A4%BF" title="त्रिकोणमिति – Νεπαλικά" lang="ne" hreflang="ne" data-title="त्रिकोणमिति" data-language-autonym="नेपाली" data-language-local-name="Νεπαλικά" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%97%E0%A5%8B%E0%A4%A8%E0%A5%8B%E0%A4%AE%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF" title="त्रिगोनोमेत्रि – Νεγουάρι" lang="new" hreflang="new" data-title="त्रिगोनोमेत्रि" data-language-autonym="नेपाल भाषा" data-language-local-name="Νεγουάρι" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Goniometrie" title="Goniometrie – Ολλανδικά" lang="nl" hreflang="nl" data-title="Goniometrie" data-language-autonym="Nederlands" data-language-local-name="Ολλανδικά" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Νορβηγικά Νινόρσκ" lang="nn" hreflang="nn" data-title="Trigonometri" data-language-autonym="Norsk nynorsk" data-language-local-name="Νορβηγικά Νινόρσκ" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Νορβηγικά Μποκμάλ" lang="nb" hreflang="nb" data-title="Trigonometri" data-language-autonym="Norsk bokmål" data-language-local-name="Νορβηγικά Μποκμάλ" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Οξιτανικά" lang="oc" hreflang="oc" data-title="Trigonometria" data-language-autonym="Occitan" data-language-local-name="Οξιτανικά" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Rogkofa" title="Rogkofa – Ορόμο" lang="om" hreflang="om" data-title="Rogkofa" data-language-autonym="Oromoo" data-language-local-name="Ορόμο" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%A4%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%95%E0%AD%8B%E0%AC%A3%E0%AC%AE%E0%AC%BF%E0%AC%A4%E0%AC%BF" title="ତ୍ରିକୋଣମିତି – Όντια" lang="or" hreflang="or" data-title="ତ୍ରିକୋଣମିତି" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Όντια" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A3%E0%A8%AE%E0%A8%BF%E0%A8%A4%E0%A9%80" title="ਤਿਕੋਣਮਿਤੀ – Παντζαπικά" lang="pa" hreflang="pa" data-title="ਤਿਕੋਣਮਿਤੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Παντζαπικά" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Trygonometria" title="Trygonometria – Πολωνικά" lang="pl" hreflang="pl" data-title="Trygonometria" data-language-autonym="Polski" data-language-local-name="Πολωνικά" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Trigonometr%C3%ACa" title="Trigonometrìa – Piedmontese" lang="pms" hreflang="pms" data-title="Trigonometrìa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%B9%D8%B1%DB%8C%DA%AF%D9%86%D9%88%D9%85%DB%8C%D9%B9%D8%B1%DB%8C" title="ٹریگنومیٹری – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ٹریگنومیٹری" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Πορτογαλικά" lang="pt" hreflang="pt" data-title="Trigonometria" data-language-autonym="Português" data-language-local-name="Πορτογαλικά" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Wamp%27artupuykama" title="Wamp'artupuykama – Κέτσουα" lang="qu" hreflang="qu" data-title="Wamp'artupuykama" data-language-autonym="Runa Simi" data-language-local-name="Κέτσουα" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Ρουμανικά" lang="ro" hreflang="ro" data-title="Trigonometrie" data-language-autonym="Română" data-language-local-name="Ρουμανικά" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.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%D1%8F" title="Тригонометрия – Ρωσικά" lang="ru" hreflang="ru" data-title="Тригонометрия" data-language-autonym="Русский" data-language-local-name="Ρωσικά" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D1%80%D1%96%D2%91%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Тріґонометрія – Rusyn" lang="rue" hreflang="rue" data-title="Тріґонометрія" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Trigunomitr%C3%ACa" title="Trigunomitrìa – Σικελικά" lang="scn" hreflang="scn" data-title="Trigunomitrìa" data-language-autonym="Sicilianu" data-language-local-name="Σικελικά" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Trigonometry" title="Trigonometry – Σκωτικά" lang="sco" hreflang="sco" data-title="Trigonometry" data-language-autonym="Scots" data-language-local-name="Σκωτικά" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Σερβοκροατικά" lang="sh" hreflang="sh" data-title="Trigonometrija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Σερβοκροατικά" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Askti%C9%A3mr" title="Asktiɣmr – Τασελχίτ" lang="shi" hreflang="shi" data-title="Asktiɣmr" data-language-autonym="Taclḥit" data-language-local-name="Τασελχίτ" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%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%BA" title="ත්‍රිකෝණමිතිය – Σινχαλεζικά" lang="si" hreflang="si" data-title="ත්‍රිකෝණමිතිය" data-language-autonym="සිංහල" data-language-local-name="Σινχαλεζικά" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Trigonometry" title="Trigonometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Trigonometry" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Σλοβακικά" lang="sk" hreflang="sk" data-title="Trigonometria" data-language-autonym="Slovenčina" data-language-local-name="Σλοβακικά" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trigonometrija" title="Trigonometrija – Σλοβενικά" lang="sl" hreflang="sl" data-title="Trigonometrija" data-language-autonym="Slovenščina" data-language-local-name="Σλοβενικά" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Pimagonyonhatu" title="Pimagonyonhatu – Σόνα" lang="sn" hreflang="sn" data-title="Pimagonyonhatu" data-language-autonym="ChiShona" data-language-local-name="Σόνα" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tirignoometeri" title="Tirignoometeri – Σομαλικά" lang="so" hreflang="so" data-title="Tirignoometeri" data-language-autonym="Soomaaliga" data-language-local-name="Σομαλικά" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Αλβανικά" lang="sq" hreflang="sq" data-title="Trigonometria" data-language-autonym="Shqip" data-language-local-name="Αλβανικά" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.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%D1%98%D0%B0" title="Тригонометрија – Σερβικά" lang="sr" hreflang="sr" data-title="Тригонометрија" data-language-autonym="Српски / srpski" data-language-local-name="Σερβικά" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Trigonometrie" title="Trigonometrie – Saterland Frisian" lang="stq" hreflang="stq" data-title="Trigonometrie" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target"><span>Seeltersk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Σουηδικά" lang="sv" hreflang="sv" data-title="Trigonometri" data-language-autonym="Svenska" data-language-local-name="Σουηδικά" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Σουαχίλι" lang="sw" hreflang="sw" data-title="Trigonometria" data-language-autonym="Kiswahili" data-language-local-name="Σουαχίλι" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%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" title="முக்கோணவியல் – Ταμιλικά" lang="ta" hreflang="ta" data-title="முக்கோணவியல்" data-language-autonym="தமிழ்" data-language-local-name="Ταμιλικά" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95%E0%B1%8B%E0%B0%A3%E0%B0%AE%E0%B0%BF%E0%B0%A4%E0%B0%BF" title="త్రికోణమితి – Τελούγκου" lang="te" hreflang="te" data-title="త్రికోణమితి" data-language-autonym="తెలుగు" data-language-local-name="Τελούγκου" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.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%D1%8F" title="Тригонометрия – Τατζικικά" lang="tg" hreflang="tg" data-title="Тригонометрия" data-language-autonym="Тоҷикӣ" data-language-local-name="Τατζικικά" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%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" title="ตรีโกณมิติ – Ταϊλανδικά" lang="th" hreflang="th" data-title="ตรีโกณมิติ" data-language-autonym="ไทย" data-language-local-name="Ταϊλανδικά" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Trigonometri%C3%BDa" title="Trigonometriýa – Τουρκμενικά" lang="tk" hreflang="tk" data-title="Trigonometriýa" data-language-autonym="Türkmençe" data-language-local-name="Τουρκμενικά" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya – Τάγκαλογκ" lang="tl" hreflang="tl" data-title="Trigonometriya" data-language-autonym="Tagalog" data-language-local-name="Τάγκαλογκ" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Trigonometri" title="Trigonometri – Τουρκικά" lang="tr" hreflang="tr" data-title="Trigonometri" data-language-autonym="Türkçe" data-language-local-name="Τουρκικά" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.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%D1%8F" title="Тригонометрия – Ταταρικά" lang="tt" hreflang="tt" data-title="Тригонометрия" data-language-autonym="Татарча / tatarça" data-language-local-name="Ταταρικά" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.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%D1%96%D1%8F" title="Тригонометрія – Ουκρανικά" lang="uk" hreflang="uk" data-title="Тригонометрія" data-language-autonym="Українська" data-language-local-name="Ουκρανικά" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB%DB%8C%D8%A7%D8%AA" title="مثلثیات – Ούρντου" lang="ur" hreflang="ur" data-title="مثلثیات" data-language-autonym="اردو" data-language-local-name="Ούρντου" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya – Ουζμπεκικά" lang="uz" hreflang="uz" data-title="Trigonometriya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Ουζμπεκικά" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Trigonometria" title="Trigonometria – Venetian" lang="vec" hreflang="vec" data-title="Trigonometria" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Trigonometrii" title="Trigonometrii – Veps" lang="vep" hreflang="vep" data-title="Trigonometrii" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C6%B0%E1%BB%A3ng_gi%C3%A1c" title="Lượng giác – Βιετναμικά" lang="vi" hreflang="vi" data-title="Lượng giác" data-language-autonym="Tiếng Việt" data-language-local-name="Βιετναμικά" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Trigonometreye" title="Trigonometreye – Βαλλωνικά" lang="wa" hreflang="wa" data-title="Trigonometreye" data-language-autonym="Walon" data-language-local-name="Βαλλωνικά" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Trigonometriya" title="Trigonometriya – Γουάραϊ" lang="war" hreflang="war" data-title="Trigonometriya" data-language-autonym="Winaray" data-language-local-name="Γουάραϊ" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%A6" title="三角学 – Κινεζικά Γου" lang="wuu" hreflang="wuu" data-title="三角学" data-language-autonym="吴语" data-language-local-name="Κινεζικά Γου" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A2%E1%83%A0%E1%83%98%E1%83%92%E1%83%9D%E1%83%9C%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90" title="ტრიგონომეტრია – Mingrelian" lang="xmf" hreflang="xmf" data-title="ტრიგონომეტრია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%98%D7%A8%D7%99%D7%92%D7%90%D7%A0%D7%90%D7%9E%D7%A2%D7%98%D7%A8%D7%99%D7%A2" title="טריגאנאמעטריע – Γίντις" lang="yi" hreflang="yi" data-title="טריגאנאמעטריע" data-language-autonym="ייִדיש" data-language-local-name="Γίντις" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/Trigonom%E1%BA%B9%CC%81tr%C3%AC" title="Trigonomẹ́trì – Γιορούμπα" lang="yo" hreflang="yo" data-title="Trigonomẹ́trì" data-language-autonym="Yorùbá" data-language-local-name="Γιορούμπα" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%A6" title="三角学 – Κινεζικά" lang="zh" hreflang="zh" data-title="三角学" data-language-autonym="中文" data-language-local-name="Κινεζικά" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Sa%E2%81%BF-kak-hoat" title="Saⁿ-kak-hoat – Minnan" lang="nan" hreflang="nan" data-title="Saⁿ-kak-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AD%B8" title="三角學 – Καντονέζικα" lang="yue" hreflang="yue" 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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Trigonometry" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q8084" title="Σύνδεσμος προς το συνδεδεμένο αντικείμενο δεδομένων [g]" accesskey="g"><span>Αντικείμενο Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Εργαλεία σελίδων"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Εμφάνιση"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div 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class="mw-indicators"> </div> <div id="siteSub" class="noprint">Από τη Βικιπαίδεια, την ελεύθερη εγκυκλοπαίδεια</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="el" dir="ltr"><p><b>Τριγωνομετρία</b> (από την ελληνική τρĩγονον "<a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο">τρίγωνο</a>" + μέτρον "<a href="/wiki/%CE%9C%CE%AD%CF%84%CF%81%CE%BF" title="Μέτρο">μέτρο</a>" ) είναι ο κλάδος των <a href="/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά">μαθηματικών</a> που ασχολείται με τη μελέτη ειδικών συναρτήσεων των γωνιών και τις εφαρμογές τους σε διάφορους υπολογισμούς, όπως στην <i>επίλυση τριγώνου</i>, δηλαδή με τον προσδιορισμό άγνωστων στοιχείων <a href="/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Τρίγωνο">τριγώνου</a>, σε συνάρτηση πλευρών και γωνιών. Η τριγωνομετρία ανάλογα του είδους των τριγώνων διακρίνεται σε <i>επίπεδη</i> και <i>σφαιρική τριγωνομετρία</i>. </p><p>Τα βασικά της τριγωνομετρίας συχνά διδάσκονται στο <a href="/wiki/%CE%A3%CF%87%CE%BF%CE%BB%CE%B5%CE%AF%CE%BF" title="Σχολείο">σχολείο</a>, είτε ως ξεχωριστό μάθημα ή ως μέρος ενός μαθήματος <a href="/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Λογισμός">λογισμού</a>. Οι τριγωνομετρικές συναρτήσεις είναι διάχυτες σε τμήματα των <a href="/wiki/%CE%9A%CE%B1%CE%B8%CE%B1%CF%81%CE%AC_%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Καθαρά μαθηματικά">καθαρών μαθηματικών</a> και των <a href="/wiki/%CE%95%CF%86%CE%B1%CF%81%CE%BC%CE%BF%CF%83%CE%BC%CE%AD%CE%BD%CE%B1_%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Εφαρμοσμένα μαθηματικά">εφαρμοσμένων μαθηματικών</a>, όπως η <a href="/wiki/%CE%91%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7_%CE%A6%CE%BF%CF%85%CF%81%CE%B9%CE%AD" title="Ανάλυση Φουριέ">ανάλυση Φουριέ</a> και την εξίσωση του κύματος, που με τη σειρά τους είναι απαραίτητα για πολλούς κλάδους της επιστήμης και της τεχνολογίας. Η σφαιρική τριγωνομετρία μελετά τρίγωνα σε <a href="/wiki/%CE%A3%CF%86%CE%B1%CE%AF%CF%81%CE%B1" title="Σφαίρα">σφαίρες</a> και επιφάνειες με σταθερή θετική καμπυλότητα στην ελλειπτική γεωμετρία. Είναι θεμελιώδους σημασίας για την <a href="/wiki/%CE%91%CF%83%CF%84%CF%81%CE%BF%CE%BD%CE%BF%CE%BC%CE%AF%CE%B1" title="Αστρονομία">αστρονομία</a> και την πλοήγηση. Η τριγωνομετρία σε επιφάνειες αρνητικής καμπυλότητας είναι μέρος της <a href="/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B2%CE%BF%CE%BB%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Υπερβολική γεωμετρία">υπερβολικής γεωμετρίας</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ιστορική_αναδρομή"><span id=".CE.99.CF.83.CF.84.CE.BF.CF.81.CE.B9.CE.BA.CE.AE_.CE.B1.CE.BD.CE.B1.CE.B4.CF.81.CE.BF.CE.BC.CE.AE"></span>Ιστορική αναδρομή</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=1" title="Επεξεργασία ενότητας: Ιστορική αναδρομή" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=1" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ιστορική αναδρομή"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ο όρος <i>τριγωνομετρία</i> καθιερώθηκε το 1595 από τον Γερμανό μαθηματικό <a href="/w/index.php?title=Bartholom%C3%A4us_Pitiscus&action=edit&redlink=1" class="new" title="Bartholomäus Pitiscus (δεν έχει γραφτεί ακόμα)">Bartholomäus Pitiscus</a> στο έργο του <i>Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus</i>. Εντούτοις η τριγωνομετρία αναπτύχθηκε και ήταν μέρος των μαθηματικών από την αρχαιότητα. Ο <a href="/wiki/%CE%91%CF%81%CE%AF%CF%83%CF%84%CE%B1%CF%81%CF%87%CE%BF%CF%82_%CE%BF_%CE%A3%CE%AC%CE%BC%CE%B9%CE%BF%CF%82" title="Αρίσταρχος ο Σάμιος">Αρίσταρχος</a> χρησιμοποίησε ορθογώνια τρίγωνα για να υπολογίσει την απόσταση της Γης από τον Ήλιο και τη Σελήνη. Οι αστρονόμοι <a href="/wiki/%CE%8A%CF%80%CF%80%CE%B1%CF%81%CF%87%CE%BF%CF%82_%CE%BF_%CE%A1%CF%8C%CE%B4%CE%B9%CE%BF%CF%82" title="Ίππαρχος ο Ρόδιος">Ίππαρχος</a> και <a href="/wiki/%CE%9A%CE%BB%CE%B1%CF%8D%CE%B4%CE%B9%CE%BF%CF%82_%CE%A0%CF%84%CE%BF%CE%BB%CE%B5%CE%BC%CE%B1%CE%AF%CE%BF%CF%82" title="Κλαύδιος Πτολεμαίος">Πτολεμαίος</a> χρησιμοποιούσαν καταλόγους που μετέτρεπαν γωνίες κύκλου σε μήκος χορδής, η γνωστή σε μας τριγωνομετρική συνάρτηση του ημίτονου. </p><p>Οι <a href="/wiki/%CE%A3%CE%BF%CF%85%CE%BC%CE%AD%CF%81%CE%B9%CE%BF%CE%B9" title="Σουμέριοι">Σουμέριοι</a> αστρονόμοι εισήγαγαν το μέτρο της γωνίας, χρησιμοποιώντας ένα διαχωρισμό του κύκλου σε 360 μοίρες. Αυτοί και οι διάδοχοί τους, οι <a href="/wiki/%CE%92%CE%B1%CE%B2%CF%85%CE%BB%CF%89%CE%BD%CE%AF%CE%B1" title="Βαβυλωνία">Βαβυλώνιοι</a> μελέτησαν τις αναλογίες των πλευρών ομοίων τριγώνων και ανακάλυψαν κάποιες ιδιότητες αυτών των αναλογιών, αλλά δεν το μετέτρεψαν σε μια συστηματική μέθοδο για την εύρεση πλευρών και γωνιών των τριγώνων. Οι <a href="/wiki/%CE%9D%CE%BF%CF%85%CE%B2%CE%AF%CE%B1" title="Νουβία">αρχαίοι Νουβίοι</a> χρησιμοποιούσαν μια παρόμοια μέθοδο. Οι <a href="/wiki/%CE%91%CF%81%CF%87%CE%B1%CE%AF%CE%B1_%CE%95%CE%BB%CE%BB%CE%AC%CE%B4%CE%B1" title="Αρχαία Ελλάδα">αρχαίοι Έλληνες</a> μετέτρεψαν την τριγωνομετρία σε μια διατεταγμένη επιστήμη. </p><p>Κλασικοί <a href="/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά">Έλληνες μαθηματικοί</a> (όπως ο <a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B7%CF%82" title="Ευκλείδης">Ευκλείδης</a> και ο <a href="/wiki/%CE%91%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82" title="Αρχιμήδης">Αρχιμήδης</a>) μελέτησαν τις ιδιότητες των <a href="/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%87%CE%BF%CF%81%CE%B4%CF%8E%CE%BD" title="Θεωρία χορδών">χορδών</a> και των χαραγμένων γωνιών σε <a href="/wiki/%CE%9A%CF%8D%CE%BA%CE%BB%CE%BF%CF%82" title="Κύκλος">κύκλους</a>, και απόδειξαν θεωρήματα που ισοδυναμούν με σύγχρονους τριγωνομετρικούς τύπους παρόλο που τα απεδείκνυαν γεωμετρικά και όχι αλγεβρικά. Ο <a href="/wiki/%CE%9A%CE%BB%CE%B1%CF%8D%CE%B4%CE%B9%CE%BF%CF%82_%CE%A0%CF%84%CE%BF%CE%BB%CE%B5%CE%BC%CE%B1%CE%AF%CE%BF%CF%82" title="Κλαύδιος Πτολεμαίος">Κλαύδιος ο Πτολεμαίος</a> διεύρυνε τις χορδές του Ίππαρχου σε ένα κύκλο στην Αλμαγέστη του. Η σύγχρονη ημιτονοειδής συνάρτηση ορίσθηκε για πρώτη φορά στη <i>Surya Siddhanta</i>, και οι ιδιότητές της ήταν τεκμηριωμένες περαιτέρω από τον Ινδό μαθηματικό και αστρονόμο του 5ου αιώνα <a href="/wiki/%CE%91%CF%81%CE%B9%CE%B1%CE%BC%CF%80%CE%AC%CF%84%CE%B1" title="Αριαμπάτα">Αριαμπάτα</a>. Τα ελληνικά και τα ινδικά αυτά έργα έχουν μεταφραστεί και επεκταθεί από Ισλαμιστές μαθηματικούς του μεσαίωνα. Μέχρι το 10ο αιώνα, οι ισλαμιστές μαθηματικοί χρησιμοποιούσαν και τις έξι τριγωνομετρικές συναρτήσεις, είχαν ταξινομημένες τις τιμές τους, και τις χρησιμοποιούσαν για τα προβλήματα στη σφαιρική γεωμετρία. Την ίδια περίπου εποχή, <a href="/wiki/%CE%9A%CE%B9%CE%BD%CE%AD%CE%B6%CE%BF%CE%B9" class="mw-redirect" title="Κινέζοι">Κινέζοι</a> μαθηματικοί ανέπτυξαν την τριγωνομετρία ανεξάρτητα, αν και δεν ήταν σημαντικό πεδίο μελέτης για αυτούς. Η γνώση των τριγωνομετρικών συναρτήσεων και μεθόδων έφτασε στην <a href="/wiki/%CE%95%CF%85%CF%81%CF%8E%CF%80%CE%B7" title="Ευρώπη">Ευρώπη</a> μέσω λατινικών μεταφράσεων των έργων των Περσών και Αράβων αστρονόμων όπως ο <a href="/wiki/%CE%91%CE%BB-%CE%9C%CF%80%CE%B1%CF%84%CE%AC%CE%BD%CE%B9" title="Αλ-Μπατάνι">Αλ-Μπατάνι</a> και <a href="/wiki/%CE%9D%CE%B1%CF%83%CE%AF%CF%81_%CE%B1%CE%BB-%CE%9D%CF%84%CE%B9%CE%BD_%CE%B1%CE%BB-%CE%A4%CE%BF%CF%85%CF%83%CE%AF" title="Νασίρ αλ-Ντιν αλ-Τουσί">Νασίρ αλ-Ντιν αλ-Τουσί</a>. Ένα από τα πρώτα έργα στην τριγωνομετρία από έναν ευρωπαίο μαθηματικό είναι το <i>De Triangulis</i> από τον Γερμανό μαθηματικό <a href="/wiki/%CE%A1%CE%B5%CE%B3%CE%B9%CE%BF%CE%BC%CE%BF%CE%BD%CF%84%CE%AC%CE%BD%CE%BF%CF%82" title="Ρεγιομοντάνος">Ρεγιομοντάνος</a> του 15ου αιώνα. Η τριγωνομετρία ήταν ακόμα τόσο λίγο γνωστή στην Ευρώπη του 16ου αιώνα που ο <a href="/wiki/%CE%9D%CE%B9%CE%BA%CF%8C%CE%BB%CE%B1%CE%BF%CF%82_%CE%9A%CE%BF%CF%80%CE%AD%CF%81%CE%BD%CE%B9%CE%BA%CE%BF%CF%82" title="Νικόλαος Κοπέρνικος">Νικόλαος Κοπέρνικος</a> αφιέρωσε δύο κεφάλαια του <i>De Revolutionibus orbium coelestium</i> για να εξηγήσει τις βασικές έννοιες. </p><p>Καθοδηγούμενη από τις απαιτήσεις της ναυσιπλοΐας και την αυξανόμενη ανάγκη για ακριβείς χάρτες των μεγάλων περιοχών, η τριγωνομετρία μεγάλωσε σε ένα σημαντικό κλάδο των μαθηματικών. Ο <a href="/w/index.php?title=Bartholomaeus_Pitiscus&action=edit&redlink=1" class="new" title="Bartholomaeus Pitiscus (δεν έχει γραφτεί ακόμα)">Bartholomaeus Pitiscus</a> ήταν ο πρώτος που χρησιμοποίησε τη λέξη, δημοσιεύοντας την <i>trigonometría</i> του το 1595. Ο <a href="/w/index.php?title=Gemma_Frisius&action=edit&redlink=1" class="new" title="Gemma Frisius (δεν έχει γραφτεί ακόμα)">Gemma Frisius</a> περιέγραψε για πρώτη φορά τη μέθοδο της τριγωνοποίησης η οποία χρησιμοποιείται ακόμα και σήμερα στη χωρομέτρηση. Ήταν ο <a href="/wiki/%CE%9B%CE%AD%CE%BF%CE%BD%CE%B1%CF%81%CE%BD%CF%84_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Λέοναρντ Όιλερ">Λέοναρντ Όιλερ</a> ο οποίος ενσωμάτωσε πλήρως τους <a href="/wiki/%CE%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Μιγαδικός αριθμός">μιγαδικούς αριθμούς</a> στην τριγωνομετρία. Τα έργα του <a href="/w/index.php?title=James_Gregory&action=edit&redlink=1" class="new" title="James Gregory (δεν έχει γραφτεί ακόμα)">James Gregory</a> τον 17ο αιώνα και του <a href="/w/index.php?title=Colin_Maclaurin&action=edit&redlink=1" class="new" title="Colin Maclaurin (δεν έχει γραφτεί ακόμα)">Colin Maclaurin</a> τον 18ο αιώνα ήταν μεγάλη επιρροή στην ανάπτυξη των τριγωνομετρικών σειρών. Επίσης, τον 18ο αιώνα, ο <a href="/w/index.php?title=Brook_Taylor&action=edit&redlink=1" class="new" title="Brook Taylor (δεν έχει γραφτεί ακόμα)">Brook Taylor</a> καθόρισε τη γενική <a href="/wiki/%CE%A3%CE%B5%CE%B9%CF%81%CE%AC_Taylor" class="mw-redirect" title="Σειρά Taylor">σειρά Taylor</a>. </p><p>Οι Άραβες υιοθέτησαν τις τριγωνομετρικές μελέτες των αρχαίων Ελλήνων και των Ινδών και ανάπτυξαν τη σφαιρική τριγωνομετρία. Οι μαθηματικοί της Ευρώπης μυήθηκαν στην τριγωνομετρία τον 15ο αιώνα, όταν την εποχή της <a href="/wiki/%CE%91%CE%BD%CE%B1%CE%B3%CE%AD%CE%BD%CE%BD%CE%B7%CF%83%CE%B7" title="Αναγέννηση">Αναγέννησης</a> ασχολήθηκαν με τον υπολογισμό βαλλιστικών τροχιών. Ο Γερμανός αστρονόμος <a href="/wiki/%CE%A1%CE%B5%CE%B3%CE%B9%CE%BF%CE%BC%CE%BF%CE%BD%CF%84%CE%AC%CE%BD%CE%BF%CF%82" title="Ρεγιομοντάνος">Ρεγιομοντάνος</a> σύνταξε μια πεντάτομη διδασκαλία της επίπεδης και σφαιρικής τριγωνομετρία με τίτλο <i>De triangulis omnimodis</i>. Σήμερα ο τρόπος γραφής των τριγωνομετρικών συναρτήσεων βασίζεται κατά μεγάλο βαθμό στα έργα του Όιλερ. </p> <div class="mw-heading mw-heading2"><h2 id="Γενικά"><span id=".CE.93.CE.B5.CE.BD.CE.B9.CE.BA.CE.AC"></span>Γενικά</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=2" title="Επεξεργασία ενότητας: Γενικά" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=2" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Γενικά"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Right_triangle_trig.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Right_triangle_trig.svg/220px-Right_triangle_trig.svg.png" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Right_triangle_trig.svg/330px-Right_triangle_trig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Right_triangle_trig.svg/440px-Right_triangle_trig.svg.png 2x" data-file-width="193" data-file-height="150" /></a><figcaption>Ένα <a href="/wiki/%CE%9F%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ορθογώνιο τρίγωνο">ορθογώνιο τρίγωνο</a>.</figcaption></figure> <p>Αν μια γωνία ενός τριγώνου είναι 90 μοίρες και μια από τις άλλες γωνίες είναι γνωστή, η τρίτη επίσης καθορίζεται, επειδή το άθροισμα των τριών γωνιών του τριγώνου είναι 180 μοίρες. Οι δύο <a href="/wiki/%CE%9F%CE%BE%CE%B5%CE%AF%CE%B1_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1" title="Οξεία γωνία">οξείες γωνίες</a> ωστόσο έχουν άθροισμα 90 μοίρες, δηλαδή είναι <a href="/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AD%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B5%CF%82" title="Συμπληρωματικές γωνίες">συμπληρωματικές γωνίες</a>. Το σχήμα ενός τριγώνου είναι απολύτως καθορισμένο από τις γωνίες, εκτός από την ομοιότητα. Όταν είναι γνωστές οι γωνίες, οι αναλογίες των πλευρών καθορίζονται ανεξαρτήτως του συνολικού μεγέθους του τριγώνου. Αν το μήκος της μίας από τις πλευρές είναι γνωστό, τότε αυτομάτως προσδιορίζονται οι άλλες δύο. Αυτές οι αναλογίες δίνονται από τις ακόλουθες τριγωνομετρικές συναρτήσεις της γνωστής γωνίας <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, όπου <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> είναι τα μήκη των πλευρών στο συνοδευτικό σχήμα: </p> <ul><li><a href="/wiki/%CE%97%CE%BC%CE%B9%CF%84%CE%BF%CE%BD%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE%CF%82_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" class="mw-redirect" title="Ημιτονοειδής συνάρτηση">Ημιτονοειδής συνάρτηση</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>), ορίζεται ως ο λόγος της απέναντι πλευράς της γωνίας προς την υποτείνουσα.</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \varphi ={\frac {\text{απέναντι πλευρά}}{\text{υποτείνουσα}}}={\frac {\alpha }{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>απέναντι πλευρά</mtext> <mtext>υποτείνουσα</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>γ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \varphi ={\frac {\text{απέναντι πλευρά}}{\text{υποτείνουσα}}}={\frac {\alpha }{\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbfe993d95d8223a55c620c478bdd9914dfab3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.192ex; height:7.843ex;" alt="{\displaystyle \sin \varphi ={\frac {\text{απέναντι πλευρά}}{\text{υποτείνουσα}}}={\frac {\alpha }{\gamma }}}"></span>.</dd></dl></dd></dl> <ul><li><a href="/w/index.php?title=%CE%A3%CF%85%CE%BD%CE%B7%CE%BC%CE%B9%CF%84%CE%BF%CE%BD%CE%B5%CE%B9%CE%B4%CE%AE%CF%82_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7&action=edit&redlink=1" class="new" title="Συνημιτονειδής συνάρτηση (δεν έχει γραφτεί ακόμα)">Συνημιτονειδής συνάρτηση</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>), ορίζεται ως ο λόγος της προσκείμενης της γωνίας προς την υποτείνουσα.</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi ={\frac {\text{προσκείμενη πλευρά}}{\text{υποτείνουσα}}}={\frac {\beta }{\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>προσκείμενη πλευρά</mtext> <mtext>υποτείνουσα</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>γ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi ={\frac {\text{προσκείμενη πλευρά}}{\text{υποτείνουσα}}}={\frac {\beta }{\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482c3bf541ff420d8eb36f868161381e3051e64a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.048ex; height:7.843ex;" alt="{\displaystyle \cos \varphi ={\frac {\text{προσκείμενη πλευρά}}{\text{υποτείνουσα}}}={\frac {\beta }{\gamma }}}"></span>.</dd></dl></dd></dl> <ul><li>Συνάρτηση της εφαπτομένης (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>), ορίζεται ως ο λόγος της απέναντι προς την προσκείμενη της γωνίας.</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \varphi ={\frac {\text{απέναντι πλευρά}}{\text{προσκείμενη πλευρά}}}={\frac {\alpha }{\beta }}={\frac {\sin \varphi }{\cos \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>απέναντι πλευρά</mtext> <mtext>προσκείμενη πλευρά</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mi>β</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \varphi ={\frac {\text{απέναντι πλευρά}}{\text{προσκείμενη πλευρά}}}={\frac {\alpha }{\beta }}={\frac {\sin \varphi }{\cos \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379da87e2133ba31228980fb26fe4f873302a987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.405ex; height:7.843ex;" alt="{\displaystyle \tan \varphi ={\frac {\text{απέναντι πλευρά}}{\text{προσκείμενη πλευρά}}}={\frac {\alpha }{\beta }}={\frac {\sin \varphi }{\cos \varphi }}}"></span>.</dd></dl></dd></dl> <p>Η <b>υποτείνουσα</b> είναι η πλευρά απέναντι από τη γωνία 90 μοιρών σε ένα ορθό τρίγωνο: είναι η μακρύτερη πλευρά του τριγώνου, και μία από τις δύο προσκείμενες πλευρές στη γωνία <i>φ</i>. Η <b>προσκείμενη πλευρά</b> είναι η πλευρά που πρόσκεινται στη γωνία <i>φ</i>. Η <b>απέναντι πλευρά</b> είναι η πλευρά που είναι απέναντι από τη γωνία <i>φ</i>. Οι κάθετοι όροι και η βάση χρησιμοποιούνται μερικές φορές για τις απέναντι και τις προσκείμενες πλευρές αντίστοιχα. </p><p>Οι αντίστροφες των συναρτήσεων αυτών ονομάζονται συντέμνουσα (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> ή <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {cosec} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {cosec} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6332f18ab20fa3bc2a7aa5bff8f325bdbe39ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.176ex; height:1.676ex;" alt="{\displaystyle \mathrm {cosec} }"></span>), τέμνουσα (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>), και συνεφαπτομένη (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>), αντίστοιχα: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc \varphi ={\frac {1}{\sin \varphi }}={\frac {\gamma }{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>csc</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>α</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc \varphi ={\frac {1}{\sin \varphi }}={\frac {\gamma }{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdf6016714d81da1e1117cb9c7396018e6fd40d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.008ex; height:5.676ex;" alt="{\displaystyle \csc \varphi ={\frac {1}{\sin \varphi }}={\frac {\gamma }{\alpha }}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec \varphi ={\frac {1}{\cos \varphi }}={\frac {\gamma }{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sec</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mi>β</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec \varphi ={\frac {1}{\cos \varphi }}={\frac {\gamma }{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f3d0efbdf0edc88ef49d3d57f98f301ffc19b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.108ex; height:5.676ex;" alt="{\displaystyle \sec \varphi ={\frac {1}{\cos \varphi }}={\frac {\gamma }{\beta }}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot \varphi ={\frac {1}{\tan \varphi }}={\frac {\cos \varphi }{\sin \varphi }}={\frac {\beta }{\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mi>α</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot \varphi ={\frac {1}{\tan \varphi }}={\frac {\cos \varphi }{\sin \varphi }}={\frac {\beta }{\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165c7f97599cebe2e8b509f2495bb3661cd03482" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.583ex; height:5.843ex;" alt="{\displaystyle \cot \varphi ={\frac {1}{\tan \varphi }}={\frac {\cos \varphi }{\sin \varphi }}={\frac {\beta }{\alpha }}}"></span>.</dd></dl> <p>Οι αντίστροφες συναρτήσεις ονομάζονται <b>τόξο ημίτονου</b>, <b>συνημίτονου</b> και <b>τόξο εφαπτομένης</b>, αντίστοιχα. Υπάρχουν αριθμητικές σχέσεις μεταξύ αυτών των συναρτήσεων οι οποίες είναι γνωστές ως τριγωνομετρικές ταυτότητες. Το συνημίτονο, η συνεφαπτομένη και η συντέμνουσα ονομάζονται έτσι επειδή είναι, αντίστοιχα, το ημίτονο, η εφαπτομένη, και η τέμνουσα της <a href="/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%BB%CE%B7%CF%81%CF%89%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AD%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B5%CF%82" title="Συμπληρωματικές γωνίες">συμπληρωματικής γωνίας</a> με τα αρχικά "συν-". </p><p>Με αυτές τις συναρτήσεις κάποιος μπορεί να απαντήσει σχεδόν σε όλες τις ερωτήσεις σχετικά με αυθαίρετα τρίγωνα χρησιμοποιώντας το νόμο των ημιτόνων και το νόμο των συνημιτόνων. Αυτοί οι νόμοι μπορούν να χρησιμοποιηθούν για να υπολογιστούν οι υπόλοιπες γωνίες και οι πλευρές οποιουδήποτε τριγώνου όταν δύο πλευρές και η περιεχόμενη γωνία ή δύο γωνίες και μία πλευρά ή τρεις πλευρές είναι γνωστές. Αυτοί οι νόμοι είναι χρήσιμοι σε όλους τους κλάδους της γεωμετρίας, αφού κάθε πολύγωνο μπορεί να περιγραφεί ως ένας πεπερασμένος συνδυασμός τριγώνων. </p> <div class="mw-heading mw-heading3"><h3 id="Η_επέκταση_των_ορισμών"><span id=".CE.97_.CE.B5.CF.80.CE.AD.CE.BA.CF.84.CE.B1.CF.83.CE.B7_.CF.84.CF.89.CE.BD_.CE.BF.CF.81.CE.B9.CF.83.CE.BC.CF.8E.CE.BD"></span>Η επέκταση των ορισμών</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=3" title="Επεξεργασία ενότητας: Η επέκταση των ορισμών" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=3" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Η επέκταση των ορισμών"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Οι παραπάνω ορισμοί ισχύουν για γωνίες μεταξύ 0 και 90 μοιρών (0 και π/2 <a href="/wiki/%CE%91%CE%BA%CF%84%CE%AF%CE%BD%CE%B9%CE%BF_(%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B1_%CE%BC%CE%AD%CF%84%CF%81%CE%B7%CF%83%CE%B7%CF%82)" title="Ακτίνιο (μονάδα μέτρησης)">ακτίνια</a>) μόνο. Χρησιμοποιώντας τον μοναδιαίο κύκλο μπορεί κανείς να τις επεκτείνει για όλες τις θετικές και αρνητικές τιμές (βλ. <a href="/wiki/%CE%A4%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%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Τριγωνομετρική συνάρτηση">τριγωνομετρική συνάρτηση</a>). Οι τριγωνομετρικές συναρτήσεις είναι <a href="/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%BF%CE%B4%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Περιοδική συνάρτηση">περιοδικές</a>, με περίοδο τις 360° μοίρες ή <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span> <a href="/wiki/%CE%91%CE%BA%CF%84%CE%AF%CE%BD%CE%B9%CE%BF_(%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B1_%CE%BC%CE%AD%CF%84%CF%81%CE%B7%CF%83%CE%B7%CF%82)" title="Ακτίνιο (μονάδα μέτρησης)">ακτίνια</a>. Αυτό σημαίνει ότι οι τιμές τους επαναλαμβάνονται σε αυτά διαστήματα. Η συνάρτηση της εφαπτομένης και της συνεφαπτομένης έχουν επίσης μια μικρότερη περίοδο, 180 μοιρών ή <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> ακτίνιων. </p><p>Οι τριγωνομετρικές συναρτήσεις μπορούν να οριστούν και με άλλους τρόπους εκτός από τoυς παραπάνω γεωμετρικούς ορισμούς, χρησιμοποιώντας εργαλεία από το <a href="/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Λογισμός">λογισμό</a> και σειρές του απειροστικού λογισμού. Αυτοί οι ορισμοί επιτρέπουν τις τριγωνομετρικές συναρτήσεις να οριστούν και για <a href="/wiki/%CE%9C%CE%B9%CE%B3%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Μιγαδικός αριθμός">μιγαδικούς αριθμούς</a>. Η σύνθετη εκθετική συνάρτηση είναι ιδιαίτερα χρήσιμη: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x+iy}=e^{x}(\cos y+i\sin y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x+iy}=e^{x}(\cos y+i\sin y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34abf830f0213aedc14a63949654d11b8dcef85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.165ex; height:3.176ex;" alt="{\displaystyle e^{x+iy}=e^{x}(\cos y+i\sin y)}"></span>.</dd></dl> <p>Δείτε τους τύπους <a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Τύπος του Όιλερ">του Όιλερ</a> και <a href="/w/index.php?title=%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CE%9C%CE%BF%CF%85%CE%AC%CE%B2%CF%81&action=edit&redlink=1" class="new" title="Τύπος Μουάβρ (δεν έχει γραφτεί ακόμα)">του ντε Μουάβρ</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Υπολογισμός_τριγωνομετρικών_συναρτήσεων"><span id=".CE.A5.CF.80.CE.BF.CE.BB.CE.BF.CE.B3.CE.B9.CF.83.CE.BC.CF.8C.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.CF.8E.CE.BD_.CF.83.CF.85.CE.BD.CE.B1.CF.81.CF.84.CE.AE.CF.83.CE.B5.CF.89.CE.BD"></span>Υπολογισμός τριγωνομετρικών συναρτήσεων</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=4" title="Επεξεργασία ενότητας: Υπολογισμός τριγωνομετρικών συναρτήσεων" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=4" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Υπολογισμός τριγωνομετρικών συναρτήσεων"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Οι τριγωνομετρικές συναρτήσεις ήταν από τις πρώτες χρήσεις των μαθηματικών πινάκων. Τέτοιοι πίνακες ενσωματώθηκαν σε μαθηματικά εγχειρίδια και οι μαθητές διδάχθηκαν να αναζητούν τις τιμές και πως να παρεμβαίνουν μεταξύ των τιμών που αναφέρονται για υψηλότερη ακρίβεια. Οι κυλιόμενοι κανόνες είχαν ειδικές κλίμακες για τριγωνομετρικές συναρτήσεις. </p><p>Σήμερα, οι επιστημονικές αριθμομηχανές έχουν κουμπιά για τον υπολογισμό των βασικών τριγωνομετρικών συναρτήσεων (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, και μερικές φορές cis και των αντιστρόφων τους). Οι περισσότερες επιτρέπουν την επιλογή της γωνίας μέτρησης: μοίρες, ακτίνια και, μερικές φορές grad. Οι περισσότερες <a href="/wiki/%CE%93%CE%BB%CF%8E%CF%83%CF%83%CE%B1_%CF%80%CF%81%CE%BF%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CE%BF%CF%8D" title="Γλώσσα προγραμματισμού">γλώσσες προγραμματισμού</a> παρέχουν βιβλιοθήκες συναρτήσεων για να υπολογίζουν τις τριγωνομετρικές συναρτήσεις. Η <a href="/wiki/%CE%9C%CE%BF%CE%BD%CE%AC%CE%B4%CE%B1_%CE%9A%CE%B9%CE%BD%CE%B7%CF%84%CE%AE%CF%82_%CE%A5%CF%80%CE%BF%CE%B4%CE%B9%CE%B1%CF%83%CF%84%CE%BF%CE%BB%CE%AE%CF%82" title="Μονάδα Κινητής Υποδιαστολής">μονάδα κινητής υποδιαστολής</a> FPU είναι ένας μικροεπεξεργαστής που υπάρχει στους περισσότερους προσωπικούς υπολογιστές, έχει ενσωματωμένες οδηγίες για τον υπολογισμό των τριγωνομετρικών συναρτήσεων. </p> <div class="mw-heading mw-heading2"><h2 id="Ταυτότητες"><span id=".CE.A4.CE.B1.CF.85.CF.84.CF.8C.CF.84.CE.B7.CF.84.CE.B5.CF.82"></span>Ταυτότητες</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=5" title="Επεξεργασία ενότητας: Ταυτότητες" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=5" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Ταυτότητες"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Σχετικές_με_το_Πυθαγόρειο_θεώρημα"><span id=".CE.A3.CF.87.CE.B5.CF.84.CE.B9.CE.BA.CE.AD.CF.82_.CE.BC.CE.B5_.CF.84.CE.BF_.CE.A0.CF.85.CE.B8.CE.B1.CE.B3.CF.8C.CF.81.CE.B5.CE.B9.CE.BF_.CE.B8.CE.B5.CF.8E.CF.81.CE.B7.CE.BC.CE.B1"></span>Σχετικές με το Πυθαγόρειο θεώρημα</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=6" title="Επεξεργασία ενότητας: Σχετικές με το Πυθαγόρειο θεώρημα" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=6" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Σχετικές με το Πυθαγόρειο θεώρημα"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Οι παρακάτω ταυτότητες σχετίζονται με το <a href="/wiki/%CE%A0%CF%85%CE%B8%CE%B1%CE%B3%CF%8C%CF%81%CE%B5%CE%B9%CE%BF_%CE%B8%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1" title="Πυθαγόρειο θεώρημα">Πυθαγόρειο θεώρημα</a>:<sup id="cite_ref-P57_1-0" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:81</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\varphi +\cos ^{2}\varphi =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\varphi +\cos ^{2}\varphi =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea2553b06db15053036ed0f1932dc29cd8e0a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.991ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}\varphi +\cos ^{2}\varphi =1}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec ^{2}\varphi -\tan ^{2}\varphi =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sec</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sec ^{2}\varphi -\tan ^{2}\varphi =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf3f32894b3064cd12a211dbead3380236e5b1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.365ex; height:3.176ex;" alt="{\displaystyle \sec ^{2}\varphi -\tan ^{2}\varphi =1}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc ^{2}\varphi -\cot ^{2}\varphi =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>csc</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <msup> <mi>cot</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \csc ^{2}\varphi -\cot ^{2}\varphi =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf40911b3ca0efa685c29effed69e2a3db316b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.105ex; height:3.176ex;" alt="{\displaystyle \csc ^{2}\varphi -\cot ^{2}\varphi =1}"></span>.</dd></dl> <p>Η δεύτερη και η τρίτη προκύπτουν από την πρώτη διαιρώντας με <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c292c6a7f127ca36fa395abaaac532d941604a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.073ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}\varphi }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7c8fb27f97644eb8cce69968251895bd2f4864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.817ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}\varphi }"></span> αντίστοιχα. </p> <div class="mw-heading mw-heading3"><h3 id="Βασικές"><span id=".CE.92.CE.B1.CF.83.CE.B9.CE.BA.CE.AD.CF.82"></span>Βασικές</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=7" title="Επεξεργασία ενότητας: Βασικές" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=7" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Βασικές"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ισχύουν οι ακόλουθες ιδιότητες για τις τριγωνομετρικές συναρτήσεις:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-P57_1-1" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 92-104">: 92-104 </span></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq \sin \varphi \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq \sin \varphi \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e46b4194c5747c279f2337830b15ce514e0764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.093ex; height:2.676ex;" alt="{\displaystyle -1\leq \sin \varphi \leq 1}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq \cos \varphi \leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\leq \cos \varphi \leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ddb8f8a5eb8e80214d979888441e41d4588623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.348ex; height:2.676ex;" alt="{\displaystyle -1\leq \cos \varphi \leq 1}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\frac {\pi }{2}}-\varphi \right)=\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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>φ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\frac {\pi }{2}}-\varphi \right)=\cos \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89742521d63ed60feb456fff9934c592161220b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.277ex; height:4.843ex;" alt="{\displaystyle \sin \left({\frac {\pi }{2}}-\varphi \right)=\cos \varphi }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \left({\frac {\pi }{2}}-\varphi \right)=\sin \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>φ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \left({\frac {\pi }{2}}-\varphi \right)=\sin \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d878648180b05e1564db6438fca0465be80687a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.277ex; height:4.843ex;" alt="{\displaystyle \cos \left({\frac {\pi }{2}}-\varphi \right)=\sin \varphi }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\pi -\varphi )=\sin \varphi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\pi -\varphi )=\sin \varphi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e92effcab5778f620f4a8fbc514a83d1fe255e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.606ex; height:2.843ex;" alt="{\displaystyle \sin(\pi -\varphi )=\sin \varphi \,}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\pi -\varphi )=-\cos \varphi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\pi -\varphi )=-\cos \varphi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5c71ad3455ce6c3d2e996609f0d362c68889c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.312ex; height:2.843ex;" alt="{\displaystyle \cos(\pi -\varphi )=-\cos \varphi \,}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\pi -\varphi )=-\tan \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\pi -\varphi )=-\tan \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63154bede6def62b74444cb0e3e787c86164b70a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.422ex; height:2.843ex;" alt="{\displaystyle \tan(\pi -\varphi )=-\tan \varphi }"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Αθροίσματος_και_διαφοράς_γωνιών"><span id=".CE.91.CE.B8.CF.81.CE.BF.CE.AF.CF.83.CE.BC.CE.B1.CF.84.CE.BF.CF.82_.CE.BA.CE.B1.CE.B9_.CE.B4.CE.B9.CE.B1.CF.86.CE.BF.CF.81.CE.AC.CF.82_.CE.B3.CF.89.CE.BD.CE.B9.CF.8E.CE.BD"></span>Αθροίσματος και διαφοράς γωνιών</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=8" title="Επεξεργασία ενότητας: Αθροίσματος και διαφοράς γωνιών" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=8" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Αθροίσματος και διαφοράς γωνιών"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Οι παρακάτω ταυτότητες σχετίζονται με το άθροισμα και την διαφορά δύο γωνιών:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-P57_1-2" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 114-123">: 114-123 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\varphi \pm \theta )=\sin \varphi \cos \theta \pm \cos \varphi \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\varphi \pm \theta )=\sin \varphi \cos \theta \pm \cos \varphi \sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a864f1927dcc734cd97d5ca1ca6d8e41db3e020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.532ex; height:2.843ex;" alt="{\displaystyle \sin(\varphi \pm \theta )=\sin \varphi \cos \theta \pm \cos \varphi \sin \theta }"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\varphi \pm \theta )=\cos \varphi \cos \theta \mp \sin \varphi \sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>∓</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\varphi \pm \theta )=\cos \varphi \cos \theta \mp \sin \varphi \sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4676eceabb4d14f5e5e7328e5be4fe7edcdd13f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.787ex; height:2.843ex;" alt="{\displaystyle \cos(\varphi \pm \theta )=\cos \varphi \cos \theta \mp \sin \varphi \sin \theta }"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(\varphi \pm \theta )={\frac {\tan \varphi \pm \tan \theta }{1\mp \tan \varphi \ \tan \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mn>1</mn> <mo>∓</mo> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(\varphi \pm \theta )={\frac {\tan \varphi \pm \tan \theta }{1\mp \tan \varphi \ \tan \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fd11970e4f4d1bc0a3a56bf8009b639dfc762e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.629ex; height:6.176ex;" alt="{\displaystyle \tan(\varphi \pm \theta )={\frac {\tan \varphi \pm \tan \theta }{1\mp \tan \varphi \ \tan \theta }}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot(\varphi \pm \theta )={\frac {\cot \varphi \ \cot \theta \mp 1}{\cot \theta \pm \cot \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> <mtext> </mtext> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>∓</mo> <mn>1</mn> </mrow> <mrow> <mi>cot</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>cot</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cot(\varphi \pm \theta )={\frac {\cot \varphi \ \cot \theta \mp 1}{\cot \theta \pm \cot \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac4bf548151eb4bd140b961c14b148da58475a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.849ex; height:6.176ex;" alt="{\displaystyle \cot(\varphi \pm \theta )={\frac {\cot \varphi \ \cot \theta \mp 1}{\cot \theta \pm \cot \varphi }}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Πολλαπλασιασμού_γωνιών"><span id=".CE.A0.CE.BF.CE.BB.CE.BB.CE.B1.CF.80.CE.BB.CE.B1.CF.83.CE.B9.CE.B1.CF.83.CE.BC.CE.BF.CF.8D_.CE.B3.CF.89.CE.BD.CE.B9.CF.8E.CE.BD"></span>Πολλαπλασιασμού γωνιών</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=9" title="Επεξεργασία ενότητας: Πολλαπλασιασμού γωνιών" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=9" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Πολλαπλασιασμού γωνιών"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Οι τριγωνομετρικοί αριθμοί της γωνίας <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4821391226411a32c838657b136fbc986d680b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.683ex; height:2.676ex;" alt="{\displaystyle 2\varphi }"></span> μπορούν να γραφτούν συναρτήσει αυτών της γωνίας <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>:<sup id="cite_ref-P57_1-3" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 124-138">: 124-138 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2\varphi )=2\sin \varphi \cdot \cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\varphi )=2\sin \varphi \cdot \cos \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ea365a11976bf1fbb765e350b9a6c6f2a10b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.456ex; height:2.843ex;" alt="{\displaystyle \sin(2\varphi )=2\sin \varphi \cdot \cos \varphi }"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\varphi )={\cos }^{2}\varphi -{\sin }^{2}\varphi =2{\cos }^{2}\varphi -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>=</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\varphi )={\cos }^{2}\varphi -{\sin }^{2}\varphi =2{\cos }^{2}\varphi -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75d967366c73216e405edc219b755657ae189be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.607ex; height:3.176ex;" alt="{\displaystyle \cos(2\varphi )={\cos }^{2}\varphi -{\sin }^{2}\varphi =2{\cos }^{2}\varphi -1}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(2\varphi )={\frac {2\tan \varphi }{1-{\tan }^{2}\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>tan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(2\varphi )={\frac {2\tan \varphi }{1-{\tan }^{2}\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0eddc2118780cf703e4519faa84147b3ee028a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.723ex; height:6.176ex;" alt="{\displaystyle \tan(2\varphi )={\frac {2\tan \varphi }{1-{\tan }^{2}\varphi }}}"></span>.</dd></dl> <p>Αντίστοιχα, για την γωνία <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39291e34614d31b498231c2f7075fe015b420112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.683ex; height:2.676ex;" alt="{\displaystyle 3\varphi }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(3\varphi )=3\sin \varphi -4{\cos }^{3}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mn>4</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(3\varphi )=3\sin \varphi -4{\cos }^{3}\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3d7abc518c55268ba11950708ea14508c6d860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.447ex; height:3.176ex;" alt="{\displaystyle \sin(3\varphi )=3\sin \varphi -4{\cos }^{3}\varphi }"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(3\varphi )=4{\cos }^{3}\varphi -3\cos \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>φ</mi> <mo>−</mo> <mn>3</mn> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(3\varphi )=4{\cos }^{3}\varphi -3\cos \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34dca55b1ea53e7928a724fcd6c3079c92f8efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.958ex; height:3.176ex;" alt="{\displaystyle \cos(3\varphi )=4{\cos }^{3}\varphi -3\cos \varphi }"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(3\varphi )={\frac {3\tan \varphi -{\tan }^{3}\varphi }{1-3\tan ^{2}\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>tan</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>φ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(3\varphi )={\frac {3\tan \varphi -{\tan }^{3}\varphi }{1-3\tan ^{2}\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d5b96c645ecbfae11c8dc2d37980c56757360e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.377ex; height:6.509ex;" alt="{\displaystyle \tan(3\varphi )={\frac {3\tan \varphi -{\tan }^{3}\varphi }{1-3\tan ^{2}\varphi }}}"></span>.</dd></dl> <p>Γενικότερα τους τριγωνομετρικούς αριθμούς της γωνίας <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01dea205937a14e59398a1ddc213ef25ddfb3265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.594ex; height:2.176ex;" alt="{\displaystyle n\cdot \varphi }"></span> για <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> <a href="/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Φυσικός αριθμός">φυσικό αριθμό</a>, μπορούμε να τις γράψουμε συναρτήσει των τριγωνομετρικών αριθμών της γωνίας <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> ως εξής: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(n\varphi )={\frac {e^{in\varphi }-e^{-in\varphi }}{2i}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \sin \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>φ</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>φ</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>φ</mi> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mi>φ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(n\varphi )={\frac {e^{in\varphi }-e^{-in\varphi }}{2i}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \sin \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbbdf4751b57e8edb71ec4449af1e83731b56b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.254ex; height:7.009ex;" alt="{\displaystyle \sin(n\varphi )={\frac {e^{in\varphi }-e^{-in\varphi }}{2i}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \sin \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(n\varphi )={\frac {e^{in\varphi }+e^{-in\varphi }}{2}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \cos \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>φ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>n</mi> <mi>φ</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>cos</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>φ</mi> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>sin</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mi>φ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(n\varphi )={\frac {e^{in\varphi }+e^{-in\varphi }}{2}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \cos \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b97f86bccbee17ef5ed25cc287bbc8498e6c1c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.765ex; height:7.009ex;" alt="{\displaystyle \cos(n\varphi )={\frac {e^{in\varphi }+e^{-in\varphi }}{2}}=\sum _{k=0}^{n}{\binom {n}{k}}\cdot {\cos }^{k}\varphi \cdot {\sin }^{n-k}\varphi \cdot \cos \left({\tfrac {1}{2}}\cdot (n-k)\cdot \pi \right)}"></span>,</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Αθροίσματος_και_διαφοράς_τριγωνομετρικών_συναρτήσεων"><span id=".CE.91.CE.B8.CF.81.CE.BF.CE.AF.CF.83.CE.BC.CE.B1.CF.84.CE.BF.CF.82_.CE.BA.CE.B1.CE.B9_.CE.B4.CE.B9.CE.B1.CF.86.CE.BF.CF.81.CE.AC.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.CF.8E.CE.BD_.CF.83.CF.85.CE.BD.CE.B1.CF.81.CF.84.CE.AE.CF.83.CE.B5.CF.89.CE.BD"></span>Αθροίσματος και διαφοράς τριγωνομετρικών συναρτήσεων</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=10" title="Επεξεργασία ενότητας: Αθροίσματος και διαφοράς τριγωνομετρικών συναρτήσεων" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=10" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Αθροίσματος και διαφοράς τριγωνομετρικών συναρτήσεων"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Για το άθροισμα και τη διαφορά των τριγωνομετρικών συναρτήσεων δύο γωνιών ισχύει ότι:<sup id="cite_ref-P57_1-4" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 139-162">: 139-162 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \varphi \pm \sin \theta =2\cdot \sin {\frac {\varphi \pm \theta }{2}}\cdot \cos {\frac {\varphi \mp \theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>±</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>∓</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \varphi \pm \sin \theta =2\cdot \sin {\frac {\varphi \pm \theta }{2}}\cdot \cos {\frac {\varphi \mp \theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ba03bb223d908e93d2970f77ab66cb5f9ff546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.871ex; height:5.509ex;" alt="{\displaystyle \sin \varphi \pm \sin \theta =2\cdot \sin {\frac {\varphi \pm \theta }{2}}\cdot \cos {\frac {\varphi \mp \theta }{2}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi +\cos \theta =2\cdot \cos {\frac {\varphi +\theta }{2}}\cdot \cos {\frac {\varphi -\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>+</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi +\cos \theta =2\cdot \cos {\frac {\varphi +\theta }{2}}\cdot \cos {\frac {\varphi -\theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071795df58ecaca6ca2d3fcbf25aac5d34410eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.637ex; height:5.509ex;" alt="{\displaystyle \cos \varphi +\cos \theta =2\cdot \cos {\frac {\varphi +\theta }{2}}\cdot \cos {\frac {\varphi -\theta }{2}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \varphi -\cos \theta =-2\cdot \sin {\frac {\varphi +\theta }{2}}\cdot \sin {\frac {\varphi -\theta }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>+</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mo>−</mo> <mi>θ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \varphi -\cos \theta =-2\cdot \sin {\frac {\varphi +\theta }{2}}\cdot \sin {\frac {\varphi -\theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49edb8b06465ff1f6420250bbc492c7cb7e59da8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.934ex; height:5.509ex;" alt="{\displaystyle \cos \varphi -\cos \theta =-2\cdot \sin {\frac {\varphi +\theta }{2}}\cdot \sin {\frac {\varphi -\theta }{2}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \varphi \pm \tan \theta ={\frac {\sin(\varphi \pm \theta )}{\cos \varphi \cdot \cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> <mo>±</mo> <mi>tan</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>±</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \varphi \pm \tan \theta ={\frac {\sin(\varphi \pm \theta )}{\cos \varphi \cdot \cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e7bbaf993db792f36b9e63503ce984eb0fe834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.165ex; height:6.343ex;" alt="{\displaystyle \tan \varphi \pm \tan \theta ={\frac {\sin(\varphi \pm \theta )}{\cos \varphi \cdot \cos \theta }}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Σε_ένα_τρίγωνο"><span id=".CE.A3.CE.B5_.CE.AD.CE.BD.CE.B1_.CF.84.CF.81.CE.AF.CE.B3.CF.89.CE.BD.CE.BF"></span>Σε ένα τρίγωνο</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=11" title="Επεξεργασία ενότητας: Σε ένα τρίγωνο" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=11" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Σε ένα τρίγωνο"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ορισμένες εξισώσεις περιλαμβανομένων των τριγωνομετρικών συναρτήσεων είναι αληθείς για όλες τις γωνίες και είναι γνωστές ως τριγωνομετρικές ταυτότητες. Ορισμένες ταυτότητες εξισώνουν μια έκφραση σε μια διαφορετική έκφραση που περιλαμβάνει τις ίδιες γωνίες. Αυτά αναφέρονται στον κατάλογο των τριγωνομετρικών ταυτοτήτων. Οι τριγωνομετρικές ταυτότητες που συνδέουν τις πλευρές και γωνίες ενός δοσμένου τριγώνου αναφέρονται παρακάτω. </p><p>Στις ακόλουθες ταυτότητες, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {\hat {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba475e78125ade3491258fde2f9efc4dc9f8afe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {A}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {\hat {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6849dc3cc33143feb098550e53a00891f402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {B}}}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {\hat {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {\hat {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9009a4994c3dac8ce2f3c83c0f205544cef28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.843ex;" alt="{\displaystyle {\rm {\hat {\Gamma }}}}"></span> είναι οι γωνίες ενός τριγώνου και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> είναι τα μήκη των πλευρών του τριγώνου απέναντι από τις αντίστοιχες γωνίες. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%CE%91%CF%81%CF%87%CE%B5%CE%AF%CE%BF:Sine_law_el.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Sine_law_el.svg/220px-Sine_law_el.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Sine_law_el.svg/330px-Sine_law_el.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Sine_law_el.svg/440px-Sine_law_el.svg.png 2x" data-file-width="207" data-file-height="184" /></a><figcaption>Τρίγωνο <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {AB\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {AB\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe8dc4aaf786375cefdcb296275bf8322d502f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.841ex; height:2.176ex;" alt="{\displaystyle {\rm {AB\Gamma }}}"></span> με τον περιγεγραμμένο του κύκλο.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Νόμος_των_ημιτόνων"><span id=".CE.9D.CF.8C.CE.BC.CE.BF.CF.82_.CF.84.CF.89.CE.BD_.CE.B7.CE.BC.CE.B9.CF.84.CF.8C.CE.BD.CF.89.CE.BD"></span>Νόμος των ημιτόνων</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=12" title="Επεξεργασία ενότητας: Νόμος των ημιτόνων" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=12" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Νόμος των ημιτόνων"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ο <a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CE%B7%CE%BC%CE%B9%CF%84%CF%8C%CE%BD%CF%89%CE%BD" title="Νόμος των ημιτόνων">νόμος των ημιτόνων</a> (επίσης γνωστός ως «κανόνας ημίτονου») για ένα τυχόν τρίγωνο είναι η εξής σχέση: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha }{\sin {\hat {\rm {A}}}}}={\frac {\beta }{\sin {\hat {\rm {B}}}}}={\frac {\gamma }{\sin {\hat {\rm {\Gamma }}}}}=2R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>γ</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\sin {\hat {\rm {A}}}}}={\frac {\beta }{\sin {\hat {\rm {B}}}}}={\frac {\gamma }{\sin {\hat {\rm {\Gamma }}}}}=2R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4007836900ee64ebb1d89eb4b179ba68bd7214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.3ex; height:6.176ex;" alt="{\displaystyle {\frac {\alpha }{\sin {\hat {\rm {A}}}}}={\frac {\beta }{\sin {\hat {\rm {B}}}}}={\frac {\gamma }{\sin {\hat {\rm {\Gamma }}}}}=2R}"></span>,</dd></dl> <p>όπου <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> είναι η <a href="/wiki/%CE%91%CE%BA%CF%84%CE%AF%CE%BD%CE%B1" class="mw-redirect" title="Ακτίνα">ακτίνα</a> του <a href="/wiki/%CE%A0%CE%B5%CF%81%CE%B9%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Περιγεγραμμένος κύκλος τριγώνου">περιγεγραμμένου κύκλου</a> του τριγώνου. </p><p>Το <a href="/wiki/%CE%95%CE%BC%CE%B2%CE%B1%CE%B4%CF%8C%CE%BD" title="Εμβαδόν">εμβαδόν</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af444c1459d0e1a3e34afa76cd4d026aabeba198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle {\rm {E}}}"></span> ενός τριγώνου δίνεται από τον παρακάτω τύπο:<sup id="cite_ref-P74_4-0" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 65">: 65 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} ={\tfrac {1}{2}}\cdot \alpha \beta \sin {\hat {\rm {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mi>α</mi> <mi>β</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} ={\tfrac {1}{2}}\cdot \alpha \beta \sin {\hat {\rm {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126595fd4f9eccc1369ecd8d94fbba5ffbffb212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.921ex; height:3.676ex;" alt="{\displaystyle \mathrm {E} ={\tfrac {1}{2}}\cdot \alpha \beta \sin {\hat {\rm {\Gamma }}}}"></span>.</dd></dl> <p>Συνδυάζοντας τους παραπάνω δύο τύπους με τον <a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%89%CF%81%CF%89%CE%BD%CE%B1" title="Τύπος του Ήρωνα">τύπο του Ήρωνα</a>, προκύπτει ότι<sup id="cite_ref-TT_5-0" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:104</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\hat {\rm {A}}}={\frac {2}{\beta \gamma }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>β</mi> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\hat {\rm {A}}}={\frac {2}{\beta \gamma }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cce38faec007ecb50a4914ac7bfe8081891939a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.394ex; height:5.843ex;" alt="{\displaystyle \sin {\hat {\rm {A}}}={\frac {2}{\beta \gamma }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\hat {\rm {B}}}={\frac {2}{\gamma \alpha }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>γ</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\hat {\rm {B}}}={\frac {2}{\gamma \alpha }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6174c2e03e7e7fbbf3a79beedbe3715b1ad392b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.452ex; height:5.676ex;" alt="{\displaystyle \sin {\hat {\rm {B}}}={\frac {2}{\gamma \alpha }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\hat {\rm {\Gamma }}}={\frac {2}{\alpha \beta }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\hat {\rm {\Gamma }}}={\frac {2}{\alpha \beta }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/339c9b1e3f40488eb7de31f6fd629a3403ae6ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.329ex; height:5.676ex;" alt="{\displaystyle \sin {\hat {\rm {\Gamma }}}={\frac {2}{\alpha \beta }}\cdot {\sqrt {\tau \cdot (\tau -\alpha )\cdot (\tau -\beta )\cdot (\tau -\gamma )}}}"></span>,</dd></dl> <p>και επίσης ότι </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {\alpha \beta \gamma }{\sqrt {(\alpha +\beta +\gamma )(\alpha -\beta +\gamma )(\alpha +\beta -\gamma )(\beta +\gamma -\alpha )}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>β</mi> <mi>γ</mi> </mrow> <msqrt> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {\alpha \beta \gamma }{\sqrt {(\alpha +\beta +\gamma )(\alpha -\beta +\gamma )(\alpha +\beta -\gamma )(\beta +\gamma -\alpha )}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6194daa6b248e8edb35cf7151ccb22cc51986190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.311ex; height:6.843ex;" alt="{\displaystyle R={\frac {\alpha \beta \gamma }{\sqrt {(\alpha +\beta +\gamma )(\alpha -\beta +\gamma )(\alpha +\beta -\gamma )(\beta +\gamma -\alpha )}}}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Νόμος_των_συνημιτόνων"><span id=".CE.9D.CF.8C.CE.BC.CE.BF.CF.82_.CF.84.CF.89.CE.BD_.CF.83.CF.85.CE.BD.CE.B7.CE.BC.CE.B9.CF.84.CF.8C.CE.BD.CF.89.CE.BD"></span>Νόμος των συνημιτόνων</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=13" title="Επεξεργασία ενότητας: Νόμος των συνημιτόνων" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=13" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Νόμος των συνημιτόνων"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ο <a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CF%83%CF%85%CE%BD%CE%B7%CE%BC%CE%B9%CF%84%CF%8C%CE%BD%CF%89%CE%BD" title="Νόμος των συνημιτόνων">νόμος των συνημιτόνων</a> είναι γενίκευση του Πυθαγορείου θεωρήματος σε τρίγωνα που δεν είναι κατά ανάγκη <a href="/wiki/%CE%9F%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF_%CF%84%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF" title="Ορθογώνιο τρίγωνο">ορθογώνια</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{2}=\alpha ^{2}+\beta ^{2}-2\alpha \beta \cos \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>α</mi> <mi>β</mi> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{2}=\alpha ^{2}+\beta ^{2}-2\alpha \beta \cos \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c522e211fa5566a5944b4f95ea1fb443bd40b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.367ex; height:3.176ex;" alt="{\displaystyle \gamma ^{2}=\alpha ^{2}+\beta ^{2}-2\alpha \beta \cos \Gamma }"></span>,</dd></dl> <p>ή ισοδύναμα: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \Gamma ={\frac {\alpha ^{2}+\beta ^{2}-\gamma ^{2}}{2\alpha \beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>α</mi> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \Gamma ={\frac {\alpha ^{2}+\beta ^{2}-\gamma ^{2}}{2\alpha \beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fe98bae1f6b97b42810ba7544159aa75c6d9d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.834ex; height:6.176ex;" alt="{\displaystyle \cos \Gamma ={\frac {\alpha ^{2}+\beta ^{2}-\gamma ^{2}}{2\alpha \beta }}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Νόμος_των_εφαπτομένων"><span id=".CE.9D.CF.8C.CE.BC.CE.BF.CF.82_.CF.84.CF.89.CE.BD_.CE.B5.CF.86.CE.B1.CF.80.CF.84.CE.BF.CE.BC.CE.AD.CE.BD.CF.89.CE.BD"></span>Νόμος των εφαπτομένων</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=14" title="Επεξεργασία ενότητας: Νόμος των εφαπτομένων" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=14" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Νόμος των εφαπτομένων"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ο <a href="/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CE%B5%CF%86%CE%B1%CF%80%CF%84%CE%BF%CE%BC%CE%AD%CE%BD%CF%89%CE%BD" title="Νόμος των εφαπτομένων">νόμος των εφαπτομένων</a> είναι η εξής σχέση: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan \left({\tfrac {1}{2}}({\hat {A}}-{\hat {B}})\right)}{\tan \left({\tfrac {1}{2}}({\hat {A}}+{\hat {B}})\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan \left({\tfrac {1}{2}}({\hat {A}}-{\hat {B}})\right)}{\tan \left({\tfrac {1}{2}}({\hat {A}}+{\hat {B}})\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad88c6cc810ece469aaae9aaf5da596c9450f5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:25.821ex; height:10.176ex;" alt="{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan \left({\tfrac {1}{2}}({\hat {A}}-{\hat {B}})\right)}{\tan \left({\tfrac {1}{2}}({\hat {A}}+{\hat {B}})\right)}}}"></span>.</dd></dl> <p>Η σχέση αυτή προκύπτει από τους <a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CE%B9_Mollweide" title="Τύποι Mollweide">τύπους Mollweide</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha +\beta }{\gamma }}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}=\cos \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> <mi>γ</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha +\beta }{\gamma }}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}=\cos \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3c7631a397964c605d7203ab0fc9a0349e1c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.068ex; height:6.343ex;" alt="{\displaystyle {\frac {\alpha +\beta }{\gamma }}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}=\cos \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}"></span>,</dd></dl> <p>και </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\alpha -\beta }{\gamma }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}=\sin \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mrow> <mi>γ</mi> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha -\beta }{\gamma }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}=\sin \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232c9efa4052afbba65a5c427a0c840b974db3c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.068ex; height:6.343ex;" alt="{\displaystyle {\frac {\alpha -\beta }{\gamma }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}=\sin \left({\tfrac {1}{2}}\cdot ({\hat {\rm {A}}}-{\hat {\rm {B}}})\right)}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Το_θεώρημα_των_προβολών"><span id=".CE.A4.CE.BF_.CE.B8.CE.B5.CF.8E.CF.81.CE.B7.CE.BC.CE.B1_.CF.84.CF.89.CE.BD_.CF.80.CF.81.CE.BF.CE.B2.CE.BF.CE.BB.CF.8E.CE.BD"></span>Το θεώρημα των προβολών</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=15" title="Επεξεργασία ενότητας: Το θεώρημα των προβολών" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=15" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Το θεώρημα των προβολών"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Το θεώρημα των προβολών δίνει ότι<sup id="cite_ref-P74_4-1" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:62</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\beta \cdot \cos {\hat {\rm {\Gamma }}}+\gamma \cdot \cos {\hat {\rm {B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>γ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\beta \cdot \cos {\hat {\rm {\Gamma }}}+\gamma \cdot \cos {\hat {\rm {B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93631fba092621e1310200c84563e8a7ab450b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:3.343ex;" alt="{\displaystyle \alpha =\beta \cdot \cos {\hat {\rm {\Gamma }}}+\gamma \cdot \cos {\hat {\rm {B}}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\gamma \cdot \cos {\hat {\rm {A}}}+\alpha \cdot \cos {\hat {\rm {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>α</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\gamma \cdot \cos {\hat {\rm {A}}}+\alpha \cdot \cos {\hat {\rm {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b32b2208a28e6219471691cd53573fdb2ec7e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.571ex; height:3.343ex;" alt="{\displaystyle \beta =\gamma \cdot \cos {\hat {\rm {A}}}+\alpha \cdot \cos {\hat {\rm {\Gamma }}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\alpha \cdot \cos {\hat {\rm {B}}}+\beta \cdot \cos {\hat {\rm {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>β</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\alpha \cdot \cos {\hat {\rm {B}}}+\beta \cdot \cos {\hat {\rm {A}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7693f9e81de6fc1104b8d26e8ec72db28dc0094f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.764ex; height:3.343ex;" alt="{\displaystyle \gamma =\alpha \cdot \cos {\hat {\rm {B}}}+\beta \cdot \cos {\hat {\rm {A}}}}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Τύπος_του_Όιλερ"><span id=".CE.A4.CF.8D.CF.80.CE.BF.CF.82_.CF.84.CE.BF.CF.85_.CE.8C.CE.B9.CE.BB.CE.B5.CF.81"></span>Τύπος του Όιλερ</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=16" title="Επεξεργασία ενότητας: Τύπος του Όιλερ" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=16" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Τύπος του Όιλερ"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ο <a href="/wiki/%CE%A4%CF%8D%CF%80%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Τύπος του Όιλερ">τύπος του Όιλερ</a> δίνει ότι <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ix}=\cos x+i\sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ix}=\cos x+i\sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.352ex; height:2.843ex;" alt="{\displaystyle e^{ix}=\cos x+i\sin x}"></span> και συνεπάγεται τις ακόλουθες αναλυτικές ταυτότητες για το ημίτονο, συνημίτονο και την εφαπτομένη όσον αφορά το <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> και τη φανταστική μονάδα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b30e164c54b1788e201501d9e510a16e1a4921b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:70.195ex; height:6.176ex;" alt="{\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Διάφορες_σχέσεις"><span id=".CE.94.CE.B9.CE.AC.CF.86.CE.BF.CF.81.CE.B5.CF.82_.CF.83.CF.87.CE.AD.CF.83.CE.B5.CE.B9.CF.82"></span>Διάφορες σχέσεις</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=17" title="Επεξεργασία ενότητας: Διάφορες σχέσεις" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=17" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Διάφορες σχέσεις"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Ισχύουν οι εξής τύποι για τις μισές γωνίες:<sup id="cite_ref-P57_1-5" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 243-246">: 243-246 </span></sup><sup id="cite_ref-P74_4-2" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 65">: 65 </span></sup><sup id="cite_ref-TT_5-1" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 104">: 104 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\beta \gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>β</mi> <mi>γ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\beta \gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495ff7ed761e79a6ad70ea2a28b7738cc7d28658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.057ex; height:7.676ex;" alt="{\displaystyle \sin {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\beta \gamma }}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \sin {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\gamma )\cdot (\tau -\alpha )}{\gamma \alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>γ</mi> <mi>α</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \sin {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\gamma )\cdot (\tau -\alpha )}{\gamma \alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5775400144ee350985fa649633c208b9d5a71342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.438ex; height:7.509ex;" alt="{\displaystyle \quad \sin {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\gamma )\cdot (\tau -\alpha )}{\gamma \alpha }}}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\alpha \beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\alpha \beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0f1ddec0b422f066e7a2a3f7eade7ba37a08b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.315ex; height:7.676ex;" alt="{\displaystyle \quad \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\alpha \beta }}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\alpha )}{\beta \gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>β</mi> <mi>γ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\alpha )}{\beta \gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68fccb834a11f0bcd42801853060eef60267691b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.556ex; height:7.676ex;" alt="{\displaystyle \cos {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\alpha )}{\beta \gamma }}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \cos {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\beta )}{\gamma \alpha }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>γ</mi> <mi>α</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \cos {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\beta )}{\gamma \alpha }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027e148302b24e0bad106e7223bc11bff27f6702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.626ex; height:7.509ex;" alt="{\displaystyle \quad \cos {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\beta )}{\gamma \alpha }}}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\gamma )}{\alpha \beta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\gamma )}{\alpha \beta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8baad22b412871c2fc40655276eb57862bd590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.363ex; height:7.676ex;" alt="{\displaystyle \quad \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {\tau \cdot (\tau -\gamma )}{\alpha \beta }}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\alpha )}}}={\frac {\rho }{\tau -\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mrow> <mi>τ</mi> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\alpha )}}}={\frac {\rho }{\tau -\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7251045fcce1e64a3e1d74662a117eb009422070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.026ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\hat {\rm {A}}}{2}}={\sqrt {\frac {(\tau -\beta )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\alpha )}}}={\frac {\rho }{\tau -\alpha }}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \tan {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\beta )}}}={\frac {\rho }{\tau -\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mrow> <mi>τ</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \tan {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\beta )}}}={\frac {\rho }{\tau -\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4257317c2fa0a601295463ed4ccf46a8dd4153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.251ex; height:7.509ex;" alt="{\displaystyle \quad \tan {\frac {\hat {\rm {B}}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\gamma )}{\tau \cdot (\tau -\beta )}}}={\frac {\rho }{\tau -\beta }}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \tan {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\tau \cdot (\tau -\gamma )}}}={\frac {\rho }{\tau -\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>τ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mrow> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \tan {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\tau \cdot (\tau -\gamma )}}}={\frac {\rho }{\tau -\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c67b5341c1c03c162bb53f89be12359b143aaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.058ex; height:7.509ex;" alt="{\displaystyle \quad \tan {\frac {\hat {\rm {\Gamma }}}{2}}={\sqrt {\frac {(\tau -\alpha )\cdot (\tau -\beta )}{\tau \cdot (\tau -\gamma )}}}={\frac {\rho }{\tau -\gamma }}}"></span>,</dd></dl></dd> <dd>όπου <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> είναι η ακτίνα του <a href="/wiki/%CE%95%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Εγγεγραμμένος κύκλος τριγώνου">εγγεγραμμένου κύκλου</a> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> η <a href="/wiki/%CE%97%CE%BC%CE%B9%CF%80%CE%B5%CF%81%CE%AF%CE%BC%CE%B5%CF%84%CF%81%CE%BF%CF%82" title="Ημιπερίμετρος">ημιπερίμετρος</a> του τριγώνου.</dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin {\hat {\rm {A}}}+\sin {\hat {\rm {B}}}+\sin {\hat {\rm {\Gamma }}}={\frac {\alpha +\beta +\gamma }{2R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin {\hat {\rm {A}}}+\sin {\hat {\rm {B}}}+\sin {\hat {\rm {\Gamma }}}={\frac {\alpha +\beta +\gamma }{2R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c86fa38690a0cfdeab4615d051a78a2d83b2e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.948ex; height:5.676ex;" alt="{\displaystyle \sin {\hat {\rm {A}}}+\sin {\hat {\rm {B}}}+\sin {\hat {\rm {\Gamma }}}={\frac {\alpha +\beta +\gamma }{2R}}}"></span>,<sup id="cite_ref-P57_1-6" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 143">: 143 </span></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2{\hat {\rm {A}}})+\sin(2{\hat {\rm {B}}})+\sin(2{\hat {\rm {\Gamma }}})=2\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2{\hat {\rm {A}}})+\sin(2{\hat {\rm {B}}})+\sin(2{\hat {\rm {\Gamma }}})=2\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412ab2d3731c5dec4dd1e535860b52822775c7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.58ex; height:3.343ex;" alt="{\displaystyle \sin(2{\hat {\rm {A}}})+\sin(2{\hat {\rm {B}}})+\sin(2{\hat {\rm {\Gamma }}})=2\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}"></span>.<sup id="cite_ref-P57_1-7" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 233">: 233 </span></sup></li> <li>Για τα γινόμενα τριγωνομετρικών συναρτήσεων ισχύουν οι εξής σχέσεις:<sup id="cite_ref-P74_4-3" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 72">: 72 </span></sup><sup id="cite_ref-TT_5-2" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 119,123">: 119,123 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\alpha +\beta +\gamma }{2R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot \cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\alpha +\beta +\gamma }{2R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722617f07046b7cc49955aedea299fb1b4a00eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.742ex; height:6.176ex;" alt="{\displaystyle 4\cdot \cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\alpha +\beta +\gamma }{2R}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho }{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho }{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9034b44ca941ae0355f7da1e28d12b7b0c1b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.976ex; height:6.176ex;" alt="{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho }{R}}}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho _{\rm {A}}}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho _{\rm {A}}}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c56f9505892967b3596e9e23739f03d3e8272c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.39ex; height:6.176ex;" alt="{\displaystyle 4\cdot \sin {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}={\frac {\rho _{\rm {A}}}{R}}}"></span>,</dd></dl></dd></dl> <ul><li>Για τα <a href="/wiki/%CE%8E%CF%88%CE%BF%CF%82_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" title="Ύψος τριγώνου">ύψη</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\rm {A}},u_{\rm {B}},u_{\rm {\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\rm {A}},u_{\rm {B}},u_{\rm {\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63fb6cfd22004f9f6a54131177054be28824590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.177ex; height:2.009ex;" alt="{\displaystyle u_{\rm {A}},u_{\rm {B}},u_{\rm {\Gamma }}}"></span> ενός τριγώνου ισχύει ότι:<sup id="cite_ref-P57_1-8" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 260">: 260 </span></sup><sup id="cite_ref-TT_5-3" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 126">: 126 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\upsilon }_{\rm {A}}=\alpha \cdot {\frac {\sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\upsilon }_{\rm {A}}=\alpha \cdot {\frac {\sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15cd759180bd976ca7546a379fbc868a95e54b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.084ex; height:6.676ex;" alt="{\displaystyle {\upsilon }_{\rm {A}}=\alpha \cdot {\frac {\sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\upsilon }_{\rm {B}}=\beta \cdot {\frac {\sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\upsilon }_{\rm {B}}=\beta \cdot {\frac {\sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc25c1b4e9edf35586eb840122013fb584f772b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.28ex; height:6.676ex;" alt="{\displaystyle \quad {\upsilon }_{\rm {B}}=\beta \cdot {\frac {\sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}}}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\upsilon }_{\rm {\Gamma }}=\gamma \cdot {\frac {\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>υ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad {\upsilon }_{\rm {\Gamma }}=\gamma \cdot {\frac {\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2008120ffd1a1ddb041a7e12349d8e646d5406e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.267ex; height:6.676ex;" alt="{\displaystyle \quad {\upsilon }_{\rm {\Gamma }}=\gamma \cdot {\frac {\sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}}}}"></span>.</dd></dl></dd></dl> <ul><li>Η ακτίνα <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> του εγγεγραμμένου κύκλου του τριγώνου, δίνεται από<sup id="cite_ref-P57_1-9" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 261-262">: 261-262 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\alpha \cdot {\frac {\sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\alpha \cdot {\frac {\sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c9268d5f67bfdb4cc1a3559b129bab3a217efb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:58.482ex; height:9.176ex;" alt="{\displaystyle \rho =\alpha \cdot {\frac {\sin {\frac {\hat {\rm {B}}}{2}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}=\beta \cdot {\frac {\sin {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}=\gamma \cdot {\frac {\sin {\frac {\hat {\rm {A}}}{2}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}"></span>,</dd></dl></dd> <dd>και από <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =(\tau -\alpha )\cdot \tan {\frac {\hat {\rm {A}}}{2}}=(\tau -\beta )\cdot \tan {\frac {\hat {\rm {B}}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\hat {\rm {\Gamma }}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =(\tau -\alpha )\cdot \tan {\frac {\hat {\rm {A}}}{2}}=(\tau -\beta )\cdot \tan {\frac {\hat {\rm {B}}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\hat {\rm {\Gamma }}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8262039ec33163db89fc76bd0b6e8bec98e0b453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.761ex; height:6.009ex;" alt="{\displaystyle \rho =(\tau -\alpha )\cdot \tan {\frac {\hat {\rm {A}}}{2}}=(\tau -\beta )\cdot \tan {\frac {\hat {\rm {B}}}{2}}=(\tau -\gamma )\cdot \tan {\frac {\hat {\rm {\Gamma }}}{2}}}"></span>.</dd></dl></dd></dl> <ul><li>Για τις <a href="/wiki/%CE%94%CE%B9%CE%AC%CE%BC%CE%B5%CF%83%CE%BF%CF%82_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Διάμεσος (γεωμετρία)">διαμέσους</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\rm {A}},\mu _{\rm {B}},\mu _{\rm {\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\rm {A}},\mu _{\rm {B}},\mu _{\rm {\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18510dcdaa9c6547f1a30f2d02621d2370b230d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.393ex; height:2.176ex;" alt="{\displaystyle \mu _{\rm {A}},\mu _{\rm {B}},\mu _{\rm {\Gamma }}}"></span> ενός τριγώνου, έχουμε ότι<sup id="cite_ref-P57_1-10" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 261-262">: 261-262 </span></sup><sup id="cite_ref-P74_4-4" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 71">: 71 </span></sup><sup id="cite_ref-TT_5-4" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 127">: 127 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\alpha }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {B}}}+2\sin ^{2}{\hat {\rm {\Gamma }}}-\sin ^{2}{\hat {\rm {A}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\alpha }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {B}}}+2\sin ^{2}{\hat {\rm {\Gamma }}}-\sin ^{2}{\hat {\rm {A}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6155eef7cc3ad24a8a1e543e03a1fbe75ff3923d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.354ex; height:6.843ex;" alt="{\displaystyle \mu _{\alpha }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {B}}}+2\sin ^{2}{\hat {\rm {\Gamma }}}-\sin ^{2}{\hat {\rm {A}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mu _{\beta }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {\Gamma }}}+2\sin ^{2}{\hat {\rm {A}}}-\sin ^{2}{\hat {\rm {B}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mu _{\beta }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {\Gamma }}}+2\sin ^{2}{\hat {\rm {A}}}-\sin ^{2}{\hat {\rm {B}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b70956c739e11e4c60ba5bf720ef63c9b9f328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.566ex; height:6.843ex;" alt="{\displaystyle \quad \mu _{\beta }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {\Gamma }}}+2\sin ^{2}{\hat {\rm {A}}}-\sin ^{2}{\hat {\rm {B}}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mu _{\gamma }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {A}}}+2\sin ^{2}{\hat {\rm {B}}}-\sin ^{2}{\hat {\rm {\Gamma }}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>4</mn> <mo>⋅</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mu _{\gamma }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {A}}}+2\sin ^{2}{\hat {\rm {B}}}-\sin ^{2}{\hat {\rm {\Gamma }}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c95ad0a4e4464543c872f5fabb809595b6c78b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.517ex; height:6.843ex;" alt="{\displaystyle \quad \mu _{\gamma }^{2}=\alpha ^{2}\cdot {\frac {2\sin ^{2}{\hat {\rm {A}}}+2\sin ^{2}{\hat {\rm {B}}}-\sin ^{2}{\hat {\rm {\Gamma }}}}{4\cdot \sin ^{2}{\hat {\rm {A}}}}}}"></span>.</dd></dl></dd></dl> <ul><li>Οι ακτίνες των <a href="/wiki/%CE%A0%CE%B1%CF%81%CE%B5%CE%B3%CE%B3%CE%B5%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%AD%CE%BD%CE%BF%CE%B9_%CE%BA%CF%8D%CE%BA%CE%BB%CE%BF%CE%B9_%CF%84%CF%81%CE%B9%CE%B3%CF%8E%CE%BD%CE%BF%CF%85" class="mw-redirect" title="Παρεγγεγραμμένοι κύκλοι τριγώνου">παρεγγεγραμμένων κύκλων</a> δίνονται από τις σχέσεις<sup id="cite_ref-P57_1-11" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 264">: 264 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f4a45c3924b81792f6c70bb4de4a54a3729983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.307ex; height:9.176ex;" alt="{\displaystyle \rho _{\rm {A}}=\alpha \cdot {\frac {\cos {\frac {\hat {\rm {B}}}{2}}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}{\cos {\frac {\hat {\rm {A}}}{2}}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \cos {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>β</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \cos {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab896643a860669a8e973abef1bc576c3d81a8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.796ex; height:9.176ex;" alt="{\displaystyle \quad \rho _{\rm {B}}=\beta \cdot {\frac {\cos {\frac {\hat {\rm {\Gamma }}}{2}}\cdot \cos {\frac {\hat {\rm {A}}}{2}}}{\cos {\frac {\hat {\rm {B}}}{2}}}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ecdbb913209d2464612f4e4bf8cd0382a16d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.404ex; height:9.176ex;" alt="{\displaystyle \quad \rho _{\rm {\Gamma }}=\gamma \cdot {\frac {\cos {\frac {\hat {\rm {A}}}{2}}\cdot \cos {\frac {\hat {\rm {B}}}{2}}}{\cos {\frac {\hat {\rm {\Gamma }}}{2}}}}}"></span>,</dd></dl></dd> <dd>και επίσης <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\rm {A}}=\tau \cdot \tan {\frac {\hat {\rm {A}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {A}}=\tau \cdot \tan {\frac {\hat {\rm {A}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18ee5e75675ee9979e1ce993339b9cbf054988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.972ex; height:6.009ex;" alt="{\displaystyle \rho _{\rm {A}}=\tau \cdot \tan {\frac {\hat {\rm {A}}}{2}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\hat {\rm {B}}}{2}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\hat {\rm {B}}}{2}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ef2b090a4c923c0dd0f6c6311dd78a7ef68d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.451ex; height:5.843ex;" alt="{\displaystyle \quad \rho _{\rm {B}}=\tau \cdot \tan {\frac {\hat {\rm {B}}}{2}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\hat {\Gamma }}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>⋅</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\hat {\Gamma }}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b69906584d5c4592723c0242363a6929be4f147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.799ex; height:5.843ex;" alt="{\displaystyle \quad \rho _{\rm {\Gamma }}=\tau \cdot \tan {\frac {\rm {\hat {\Gamma }}}{2}}}"></span>.</dd></dl></dd></dl> <ul><li>Τα μήκη των <a href="/wiki/%CE%94%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%82_%CE%B3%CF%89%CE%BD%CE%AF%CE%B1%CF%82" title="Διχοτόμος γωνίας">εσωτερικών διχοτόμων</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}},\delta _{\rm {B}},\delta _{\rm {\Gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}},\delta _{\rm {B}},\delta _{\rm {\Gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c234765ce049424072e32c93d0e372779baca5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.285ex; height:2.676ex;" alt="{\displaystyle \delta _{\rm {A}},\delta _{\rm {B}},\delta _{\rm {\Gamma }}}"></span> δίνονται από τις σχέσεις<sup id="cite_ref-P57_1-12" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 265-266">: 265-266 </span></sup><sup id="cite_ref-P74_4-5" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 69">: 69 </span></sup><sup id="cite_ref-TT_5-5" class="reference"><a href="#cite_note-TT-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 128">: 128 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}}={\frac {2\beta \gamma }{\beta +\gamma }}\cdot \cos {\frac {\hat {\rm {A}}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>β</mi> <mi>γ</mi> </mrow> <mrow> <mi>β</mi> <mo>+</mo> <mi>γ</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}}={\frac {2\beta \gamma }{\beta +\gamma }}\cdot \cos {\frac {\hat {\rm {A}}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6719aee435d8affb362b1eb2c163d2f54ae3aa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.27ex; height:6.676ex;" alt="{\displaystyle \delta _{\rm {A}}={\frac {2\beta \gamma }{\beta +\gamma }}\cdot \cos {\frac {\hat {\rm {A}}}{2}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {B}}={\frac {2\gamma \alpha }{\gamma +\alpha }}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>γ</mi> <mi>α</mi> </mrow> <mrow> <mi>γ</mi> <mo>+</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {B}}={\frac {2\gamma \alpha }{\gamma +\alpha }}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df8dacc2f60161fb931d68368bc744eda3ee98eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.257ex; height:6.343ex;" alt="{\displaystyle \quad \delta _{\rm {B}}={\frac {2\gamma \alpha }{\gamma +\alpha }}\cdot \cos {\frac {\hat {\rm {B}}}{2}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {2\alpha \beta }{\alpha +\beta }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {2\alpha \beta }{\alpha +\beta }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6789ff45c9cce8658e8684d22c39b8a95eabb866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.675ex; height:6.343ex;" alt="{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {2\alpha \beta }{\alpha +\beta }}\cdot \cos {\frac {\hat {\rm {\Gamma }}}{2}}}"></span>,</dd></dl></dd> <dd>και επίσης <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}}={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\rm {A}}\cdot \cos {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}}={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\rm {A}}\cdot \cos {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7c957df11158ac6334ca63535cd8f1897ed547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.9ex; height:8.009ex;" alt="{\displaystyle \delta _{\rm {A}}={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\rm {A}}\cdot \cos {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {B}}={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \cos {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {B}}={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \cos {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3a5664b09cd6c2d9d5b178393986626888b405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:25.448ex; height:8.343ex;" alt="{\displaystyle \quad \delta _{\rm {B}}={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \cos {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \cos {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \cos {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e390298aa3431a17ac95a364589984397248ccd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:23.044ex; height:8.509ex;" alt="{\displaystyle \quad \delta _{\rm {\Gamma }}={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \cos {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}"></span>.</dd></dl></dd></dl> <ul><li>Τα μήκη των <a href="/wiki/%CE%95%CE%BE%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CF%87%CE%BF%CF%84%CF%8C%CE%BC%CE%BF%CF%82" class="mw-redirect" title="Εξωτερική διχοτόμος">εξωτερικών διχοτόμων</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}}',\delta _{\rm {B}}',\delta _{\rm {\Gamma }}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}}',\delta _{\rm {B}}',\delta _{\rm {\Gamma }}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a122bda89059dc34a1d271ee2112d555aac4221d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.285ex; height:3.009ex;" alt="{\displaystyle \delta _{\rm {A}}',\delta _{\rm {B}}',\delta _{\rm {\Gamma }}'}"></span> δίνονται από τις σχέσεις<sup id="cite_ref-P57_1-13" class="reference"><a href="#cite_note-P57-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 266-267">: 266-267 </span></sup><sup id="cite_ref-P74_4-6" class="reference"><a href="#cite_note-P74-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Σελίδα: 70">: 70 </span></sup></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}}'={\frac {2\beta \gamma }{|\beta -\gamma |}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>β</mi> <mi>γ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mo>−</mo> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}}'={\frac {2\beta \gamma }{|\beta -\gamma |}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf57b228e0d9e3e4ceadbfef55112d04b07d7c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.661ex; height:6.843ex;" alt="{\displaystyle \delta _{\rm {A}}'={\frac {2\beta \gamma }{|\beta -\gamma |}}\cdot \sin {\frac {\hat {\rm {A}}}{2}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {B}}'={\frac {2\gamma \alpha }{|\gamma -\alpha |}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>γ</mi> <mi>α</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>γ</mi> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {B}}'={\frac {2\gamma \alpha }{|\gamma -\alpha |}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/559b9ed6902e197cc6594928402ce6ba9d2f863d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.296ex; height:6.676ex;" alt="{\displaystyle \quad \delta _{\rm {B}}'={\frac {2\gamma \alpha }{|\gamma -\alpha |}}\cdot \sin {\frac {\hat {\rm {B}}}{2}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {2\alpha \beta }{|\alpha -\beta |}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>α</mi> <mi>β</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {2\alpha \beta }{|\alpha -\beta |}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3694f8f33ff6dd891f86a86c848c30483b6a4824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.714ex; height:6.676ex;" alt="{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {2\alpha \beta }{|\alpha -\beta |}}\cdot \sin {\frac {\hat {\rm {\Gamma }}}{2}}}"></span>.</dd></dl></dd> <dd>και επίσης <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\rm {A}}'={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}\cdot \sin {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\rm {A}}'={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}\cdot \sin {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c51e79d9f91aa8d7e2fdbd3981f3d8807524554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.861ex; height:8.009ex;" alt="{\displaystyle \delta _{\rm {A}}'={\frac {\alpha \cdot \sin {\hat {\rm {B}}}\cdot \sin {\hat {\rm {\Gamma }}}}{\sin {\hat {\rm {A}}}\cdot \sin {\frac {{\hat {\rm {B}}}-{\hat {\rm {\Gamma }}}}{2}}}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {B}}'={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \sin {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {B}}'={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \sin {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8d057a0f94645d746d2ef137aa6691317912b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:25.38ex; height:8.343ex;" alt="{\displaystyle \quad \delta _{\rm {B}}'={\frac {\beta \cdot \sin {\hat {\rm {\Gamma }}}\cdot \sin {\hat {\rm {A}}}}{\sin {\hat {\rm {B}}}\cdot \sin {\frac {{\hat {\rm {\Gamma }}}-{\hat {\rm {A}}}}{2}}}}\quad }"></span> και <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \sin {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \sin {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be1ac7ff6aea82ebac6470b8f71e5bbf5092caa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:23.044ex; height:8.509ex;" alt="{\displaystyle \quad \delta _{\rm {\Gamma }}'={\frac {\gamma \cdot \sin {\hat {\rm {A}}}\cdot \sin {\hat {\rm {B}}}}{\sin {\hat {\rm {\Gamma }}}\cdot \sin {\frac {{\hat {\rm {A}}}-{\hat {\rm {B}}}}{2}}}}}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Σφαιρική_τριγωνομετρία"><span id=".CE.A3.CF.86.CE.B1.CE.B9.CF.81.CE.B9.CE.BA.CE.AE_.CF.84.CF.81.CE.B9.CE.B3.CF.89.CE.BD.CE.BF.CE.BC.CE.B5.CF.84.CF.81.CE.AF.CE.B1"></span>Σφαιρική τριγωνομετρία</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=18" title="Επεξεργασία ενότητας: Σφαιρική τριγωνομετρία" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=18" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Σφαιρική τριγωνομετρία"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Η σφαιρική τριγωνομετρία αποτελεί εν μέρει αντικείμενο της <a href="/wiki/%CE%9F%CF%85%CF%81%CE%AC%CE%BD%CE%B9%CE%B1_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Ουράνια μηχανική">ουράνιας μηχανικής</a> στην <a href="/wiki/%CE%91%CF%83%CF%84%CF%81%CE%BF%CE%BD%CE%BF%CE%BC%CE%AF%CE%B1" title="Αστρονομία">αστρονομία</a> και αφορά στην επίλυση σφαιρικών τριγώνων. </p> <div class="mw-heading mw-heading2"><h2 id="Εφαρμογές_της_τριγωνομετρίας"><span id=".CE.95.CF.86.CE.B1.CF.81.CE.BC.CE.BF.CE.B3.CE.AD.CF.82_.CF.84.CE.B7.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.AF.CE.B1.CF.82"></span>Εφαρμογές της τριγωνομετρίας</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=19" title="Επεξεργασία ενότητας: Εφαρμογές της τριγωνομετρίας" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=19" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Εφαρμογές της τριγωνομετρίας"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Η τριγωνομετρία και οι τριγωνομετρικές συναρτήσεις βρίσκουν εφαρμογές σε πολλούς τομείς. Για παράδειγμα, η τεχνική του τριγωνισμού χρησιμοποιείται στην <a href="/wiki/%CE%91%CF%83%CF%84%CF%81%CE%BF%CE%BD%CE%BF%CE%BC%CE%AF%CE%B1" title="Αστρονομία">αστρονομία</a> για να μετρήσει την απόσταση σε κοντινά αστέρια, στη <a href="/wiki/%CE%93%CE%B5%CF%89%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Γεωγραφία">γεωγραφία</a> για τη μέτρηση αποστάσεων μεταξύ ορόσημων, και στα συστήματα δορυφορικής πλοήγησης. Οι συναρτήσεις ημίτονου και συνημίτονου είναι θεμελιώδους σημασίας στη θεωρία των περιοδικών συναρτήσεων όπως αυτές που περιγράφουν τα κύματα ήχου και το φωτός. </p><p>Τα επιστημονικά πεδία που χρησιμοποιούν την τριγωνομετρία ή τις τριγωνομετρικές συναρτήσεις συμπεριλαμβάνουν το πεδίο της <a href="/wiki/%CE%91%CF%83%CF%84%CF%81%CE%BF%CE%BD%CE%BF%CE%BC%CE%AF%CE%B1" title="Αστρονομία">αστρονομίας</a> (ειδικά για τον εντοπισμό θέσης των ουράνιων αντικειμένων, στην οποία η σφαιρική τριγωνομετρία είναι απαραίτητη) και ως εκ τούτου στην <a href="/wiki/%CE%9D%CE%B1%CF%85%CF%83%CE%B9%CF%80%CE%BB%CE%BF%CE%90%CE%B1" title="Ναυσιπλοΐα">πλοήγηση</a> (στους ωκεανούς, σε αεροσκάφη, και στο διάστημα), θεωρία της μουσικής, σύνθεση ήχου, <a href="/wiki/%CE%91%CE%BA%CE%BF%CF%85%CF%83%CF%84%CE%B9%CE%BA%CE%AE" title="Ακουστική">ακουστική</a>, <a href="/wiki/%CE%9F%CF%80%CF%84%CE%B9%CE%BA%CE%AE" title="Οπτική">οπτική</a>, ανάλυση των χρηματοπιστωτικών αγορών, <a href="/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CF%80%CE%B9%CE%B8%CE%B1%CE%BD%CE%BF%CF%84%CE%AE%CF%84%CF%89%CE%BD" title="Θεωρία πιθανοτήτων">θεωρία πιθανοτήτων</a>, <a href="/wiki/%CE%A3%CF%84%CE%B1%CF%84%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE" title="Στατιστική">στατιστική</a>, <a href="/wiki/%CE%92%CE%B9%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Βιολογία">βιολογία</a>, <a href="/wiki/%CE%91%CE%BA%CF%84%CE%B9%CE%BD%CE%BF%CE%B4%CE%B9%CE%B1%CE%B3%CE%BD%CF%89%CF%83%CF%84%CE%B9%CE%BA%CE%AE" title="Ακτινοδιαγνωστική">ακτινοδιαγνωστική</a> (<a href="/wiki/%CE%91%CE%BA%CF%84%CE%B9%CE%BD%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Ακτινογραφία">ακτινογραφία</a> και υπερηχογράφημα), <a href="/wiki/%CE%A6%CE%B1%CF%81%CE%BC%CE%B1%CE%BA%CE%B5%CF%85%CF%84%CE%B9%CE%BA%CE%AE" title="Φαρμακευτική">φαρμακευτική</a>, <a href="/wiki/%CE%A7%CE%B7%CE%BC%CE%B5%CE%AF%CE%B1" title="Χημεία">χημεία</a>, <a href="/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD" title="Θεωρία αριθμών">θεωρία αριθμών</a> (και ως εκ τούτου, <a href="/wiki/%CE%9A%CF%81%CF%85%CF%80%CF%84%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Κρυπτογραφία">κρυπτογραφία</a>), <a href="/wiki/%CE%A3%CE%B5%CE%B9%CF%83%CE%BC%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Σεισμολογία">σεισμολογία</a>, <a href="/wiki/%CE%9C%CE%B5%CF%84%CE%B5%CF%89%CF%81%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Μετεωρολογία">μετεωρολογία</a>, <a href="/wiki/%CE%A9%CE%BA%CE%B5%CE%B1%CE%BD%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Ωκεανογραφία">ωκεανογραφία</a>, πολλές <a href="/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CE%AD%CF%82_%CE%B5%CF%80%CE%B9%CF%83%CF%84%CE%AE%CE%BC%CE%B5%CF%82" title="Φυσικές επιστήμες">φυσικές επιστήμες</a>, <a href="/wiki/%CE%A4%CE%BF%CF%80%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Τοπογραφία">τοπογραφία</a> και <a href="/wiki/%CE%93%CE%B5%CF%89%CE%B4%CE%B1%CE%B9%CF%83%CE%AF%CE%B1" title="Γεωδαισία">γεωδαισία</a>, την <a href="/wiki/%CE%91%CF%81%CF%87%CE%B9%CF%84%CE%B5%CE%BA%CF%84%CE%BF%CE%BD%CE%B9%CE%BA%CE%AE" title="Αρχιτεκτονική">αρχιτεκτονική</a>, <a href="/wiki/%CE%A6%CF%89%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CE%AE" title="Φωνητική">φωνητική</a>, <a href="/wiki/%CE%9F%CE%B9%CE%BA%CE%BF%CE%BD%CE%BF%CE%BC%CE%AF%CE%B1" title="Οικονομία">οικονομία</a>, <a href="/wiki/%CE%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%BF%CE%BD%CE%B9%CE%BA%CE%AE" title="Ηλεκτρονική">ηλεκτρονική</a>, <a href="/wiki/%CE%9C%CE%B7%CF%87%CE%B1%CE%BD%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Μηχανολογία">μηχανολογία</a>, <a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B1%CF%83%CE%BA%CE%B5%CF%85%CE%AE" title="Κατασκευή">κατασκευές</a>, <a href="/wiki/%CE%93%CF%81%CE%B1%CF%86%CE%B9%CE%BA%CE%AC_%CF%85%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CF%84%CF%8E%CE%BD" title="Γραφικά υπολογιστών">γραφικά υπολογιστών</a>, <a href="/wiki/%CE%A7%CE%B1%CF%81%CF%84%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Χαρτογραφία">χαρτογραφία</a>, <a href="/wiki/%CE%9A%CF%81%CF%85%CF%83%CF%84%CE%B1%CE%BB%CE%BB%CE%BF%CE%B3%CF%81%CE%B1%CF%86%CE%AF%CE%B1" title="Κρυσταλλογραφία">κρυσταλλογραφία</a> και την ανάπτυξη παιχνιδιών. </p> <div class="mw-heading mw-heading2"><h2 id="Δείτε_επίσης"><span id=".CE.94.CE.B5.CE.AF.CF.84.CE.B5_.CE.B5.CF.80.CE.AF.CF.83.CE.B7.CF.82"></span>Δείτε επίσης</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=20" title="Επεξεργασία ενότητας: Δείτε επίσης" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=20" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Δείτε επίσης"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%CE%A4%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%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Τριγωνομετρική συνάρτηση">Τριγωνομετρική συνάρτηση</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Παραπομπές"><span id=".CE.A0.CE.B1.CF.81.CE.B1.CF.80.CE.BF.CE.BC.CF.80.CE.AD.CF.82"></span>Παραπομπές</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=21" title="Επεξεργασία ενότητας: Παραπομπές" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=21" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Παραπομπές"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-P57-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-P57_1-0">1,00</a></sup> <sup><a href="#cite_ref-P57_1-1">1,01</a></sup> <sup><a href="#cite_ref-P57_1-2">1,02</a></sup> <sup><a href="#cite_ref-P57_1-3">1,03</a></sup> <sup><a href="#cite_ref-P57_1-4">1,04</a></sup> <sup><a href="#cite_ref-P57_1-5">1,05</a></sup> <sup><a href="#cite_ref-P57_1-6">1,06</a></sup> <sup><a href="#cite_ref-P57_1-7">1,07</a></sup> <sup><a href="#cite_ref-P57_1-8">1,08</a></sup> <sup><a href="#cite_ref-P57_1-9">1,09</a></sup> <sup><a href="#cite_ref-P57_1-10">1,10</a></sup> <sup><a href="#cite_ref-P57_1-11">1,11</a></sup> <sup><a href="#cite_ref-P57_1-12">1,12</a></sup> <sup><a href="#cite_ref-P57_1-13">1,13</a></sup></span> <span class="reference-text"><cite class="citation book">Τόγκας, Πέτρος Γ. (1957). <i>Ευθύγραμμος τριγωνομετρία</i>. Αθήνα: Εκδοτικός Οίκος Πέτρου Γ. Τόγκα.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%95%CF%85%CE%B8%CF%8D%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%BF%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%AF%CE%B1&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%95%CE%BA%CE%B4%CE%BF%CF%84%CE%B9%CE%BA%CF%8C%CF%82+%CE%9F%CE%AF%CE%BA%CE%BF%CF%82+%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%85+%CE%93.+%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1&rft.date=1957&rft.aulast=%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1%CF%82&rft.aufirst=%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%82+%CE%93.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Αργυρόπουλος, Η.· Βλάμος, Π.· Κατσούλης, Γ.· Μαρκάτσης, Σ.· Σιδερης, Π. (2002). <i>Ευκλείδεια Γεωμετρία (Α΄ και Β΄Λυκείου)</i>. ΑΘΗΝΑ: ΟΕΔΒ. σελ. 192.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1+%CE%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1+%28%CE%91%CE%84+%CE%BA%CE%B1%CE%B9+%CE%92%CE%84%CE%9B%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85%29&rft.place=%CE%91%CE%98%CE%97%CE%9D%CE%91&rft.pages=192&rft.pub=%CE%9F%CE%95%CE%94%CE%92&rft.date=2002&rft.aulast=%CE%91%CF%81%CE%B3%CF%85%CF%81%CF%8C%CF%80%CE%BF%CF%85%CE%BB%CE%BF%CF%82&rft.aufirst=%CE%97.&rft.au=%CE%92%CE%BB%CE%AC%CE%BC%CE%BF%CF%82%2C+%CE%A0.&rft.au=%CE%9A%CE%B1%CF%84%CF%83%CE%BF%CF%8D%CE%BB%CE%B7%CF%82%2C+%CE%93.&rft.au=%CE%9C%CE%B1%CF%81%CE%BA%CE%AC%CF%84%CF%83%CE%B7%CF%82%2C+%CE%A3.&rft.au=%CE%A3%CE%B9%CE%B4%CE%B5%CF%81%CE%B7%CF%82%2C+%CE%A0.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation book">Ανδρεαδακης, Σ.· Κατσαργυρης, Β.· Παπασταυριδης, Σ.· Πολυζος, Γ.· Σβερκος, Α. (1998). <i>Άλγεβρα Β΄ Γενικού Λυκείου</i>. ΑΘΗΝΑ: ΟΕΔΒ. σελ. 26-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1+%CE%92%CE%84+%CE%93%CE%B5%CE%BD%CE%B9%CE%BA%CE%BF%CF%8D+%CE%9B%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85&rft.place=%CE%91%CE%98%CE%97%CE%9D%CE%91&rft.pages=26-27&rft.pub=%CE%9F%CE%95%CE%94%CE%92&rft.date=1998&rft.aulast=%CE%91%CE%BD%CE%B4%CF%81%CE%B5%CE%B1%CE%B4%CE%B1%CE%BA%CE%B7%CF%82&rft.aufirst=%CE%A3.&rft.au=%CE%9A%CE%B1%CF%84%CF%83%CE%B1%CF%81%CE%B3%CF%85%CF%81%CE%B7%CF%82%2C+%CE%92.&rft.au=%CE%A0%CE%B1%CF%80%CE%B1%CF%83%CF%84%CE%B1%CF%85%CF%81%CE%B9%CE%B4%CE%B7%CF%82%2C+%CE%A3.&rft.au=%CE%A0%CE%BF%CE%BB%CF%85%CE%B6%CE%BF%CF%82%2C+%CE%93.&rft.au=%CE%A3%CE%B2%CE%B5%CF%81%CE%BA%CE%BF%CF%82%2C+%CE%91.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-P74-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-P74_4-0">4,0</a></sup> <sup><a href="#cite_ref-P74_4-1">4,1</a></sup> <sup><a href="#cite_ref-P74_4-2">4,2</a></sup> <sup><a href="#cite_ref-P74_4-3">4,3</a></sup> <sup><a href="#cite_ref-P74_4-4">4,4</a></sup> <sup><a href="#cite_ref-P74_4-5">4,5</a></sup> <sup><a href="#cite_ref-P74_4-6">4,6</a></sup></span> <span class="reference-text"><cite class="citation book">Παπατριανταφύλλου, Ε. (1974). <i>Μαθηματικά ΣΤ' Γυμνασίου Θετικής Κατευθύνσεως: Τριγωνομετρία</i>. Αθήνα: Οργανισμός Εκδόσεων Διδακτικών Βιβλίων.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC+%CE%A3%CE%A4%27+%CE%93%CF%85%CE%BC%CE%BD%CE%B1%CF%83%CE%AF%CE%BF%CF%85+%CE%98%CE%B5%CF%84%CE%B9%CE%BA%CE%AE%CF%82+%CE%9A%CE%B1%CF%84%CE%B5%CF%85%CE%B8%CF%8D%CE%BD%CF%83%CE%B5%CF%89%CF%82%3A+%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%9F%CF%81%CE%B3%CE%B1%CE%BD%CE%B9%CF%83%CE%BC%CF%8C%CF%82+%CE%95%CE%BA%CE%B4%CF%8C%CF%83%CE%B5%CF%89%CE%BD+%CE%94%CE%B9%CE%B4%CE%B1%CE%BA%CF%84%CE%B9%CE%BA%CF%8E%CE%BD+%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CF%89%CE%BD&rft.date=1974&rft.aulast=%CE%A0%CE%B1%CF%80%CE%B1%CF%84%CF%81%CE%B9%CE%B1%CE%BD%CF%84%CE%B1%CF%86%CF%8D%CE%BB%CE%BB%CE%BF%CF%85&rft.aufirst=%CE%95.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-TT-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-TT_5-0">5,0</a></sup> <sup><a href="#cite_ref-TT_5-1">5,1</a></sup> <sup><a href="#cite_ref-TT_5-2">5,2</a></sup> <sup><a href="#cite_ref-TT_5-3">5,3</a></sup> <sup><a href="#cite_ref-TT_5-4">5,4</a></sup> <sup><a href="#cite_ref-TT_5-5">5,5</a></sup></span> <span class="reference-text"><cite class="citation book">Τόγκας, Πέτρος Γ. <i>Ασκήσεις και προβλήματα τριγωνομετρίας</i>. Αθήνα: Εκδοτικός Οίκος Πέτρου Γ. Τόγκα.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%CE%91%CF%83%CE%BA%CE%AE%CF%83%CE%B5%CE%B9%CF%82+%CE%BA%CE%B1%CE%B9+%CF%80%CF%81%CE%BF%CE%B2%CE%BB%CE%AE%CE%BC%CE%B1%CF%84%CE%B1+%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1%CF%82&rft.place=%CE%91%CE%B8%CE%AE%CE%BD%CE%B1&rft.pub=%CE%95%CE%BA%CE%B4%CE%BF%CF%84%CE%B9%CE%BA%CF%8C%CF%82+%CE%9F%CE%AF%CE%BA%CE%BF%CF%82+%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%85+%CE%93.+%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1&rft.aulast=%CE%A4%CF%8C%CE%B3%CE%BA%CE%B1%CF%82&rft.aufirst=%CE%A0%CE%AD%CF%84%CF%81%CE%BF%CF%82+%CE%93.&rfr_id=info%3Asid%2Fel.wikipedia.org%3A%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Εξωτερικοί_σύνδεσμοι"><span id=".CE.95.CE.BE.CF.89.CF.84.CE.B5.CF.81.CE.B9.CE.BA.CE.BF.CE.AF_.CF.83.CF.8D.CE.BD.CE.B4.CE.B5.CF.83.CE.BC.CE.BF.CE.B9"></span>Εξωτερικοί σύνδεσμοι</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&veaction=edit&section=22" title="Επεξεργασία ενότητας: Εξωτερικοί σύνδεσμοι" class="mw-editsection-visualeditor"><span>Επεξεργασία</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&section=22" title="Επεξεργαστείτε τον πηγαίο κώδικα της ενότητας: Εξωτερικοί σύνδεσμοι"><span>επεξεργασία κώδικα</span></a><span class="mw-editsection-bracket">]</span></span></div> <table role="presentation" class="noprint mbox-small" style="border:1px solid var(--border-color-base, #a2a9b1); background-color:var(--background-color-notice-subtle, #f8f9fa); color:inherit;"> <tbody><tr> <td class="mbox-image"><figure class="mw-halign-none" typeof="mw:File"><span title="wiktionary logo"><img alt="wiktionary logo" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/40px-Wiktionary-logo.svg.png" decoding="async" width="40" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/60px-Wiktionary-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/80px-Wiktionary-logo.svg.png 2x" data-file-width="370" data-file-height="350" /></span><figcaption>wiktionary logo</figcaption></figure></td> <td class="mbox-text plainlist" style="">Το <i><a href="https://el.wiktionary.org/wiki/el:" class="extiw" title="wikt:el:">Βικιλεξικό</a></i> έχει σχετικό λήμμα:<br />  <b><a href="https://el.wiktionary.org/wiki/el:%CF%84%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" class="extiw" title="wikt:el:τριγωνομετρία">τριγωνομετρία </a></b></td> </tr> </tbody></table> <table role="presentation" class="noprint mbox-small" style="border:1px solid var(--border-color-base, #a2a9b1); background-color:var(--background-color-notice-subtle, #f8f9fa); color:inherit;"> <tbody><tr> <td class="mbox-image"><figure class="mw-halign-none" typeof="mw:File"><span title="Commons logo"><img alt="Commons logo" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png" decoding="async" width="40" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/80px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span><figcaption>Commons logo</figcaption></figure></td> <td class="mbox-text plainlist" style="">Τα <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a> έχουν πολυμέσα σχετικά με το θέμα<br />  <b><a href="https://commons.wikimedia.org/wiki/Category:Trigonometry" class="extiw" title="commons:Category:Trigonometry"> Τριγωνομετρία</a></b></td> </tr> </tbody></table> <ul><li><a rel="nofollow" class="external free" href="http://portal.survey.ntua.gr/main/courses/hisatgeodesy/geoastro/book/GeoAstro_chapter11.pdf">http://portal.survey.ntua.gr/main/courses/hisatgeodesy/geoastro/book/GeoAstro_chapter11.pdf</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130418203343/http://portal.survey.ntua.gr/main/courses/hisatgeodesy/geoastro/book/GeoAstro_chapter11.pdf">Αρχειοθετήθηκε</a> 2013-04-18 στο <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li></ul> <p><br /> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r10387572">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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title="Γεωμετρία">γεωμετρίας</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=%CE%91%CF%80%CF%8C%CE%BB%CF%85%CF%84%CE%B7_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&redlink=1" class="new" title="Απόλυτη γεωμετρία (δεν έχει γραφτεί ακόμα)">απόλυτες γεωμετρίες</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%CE%95%CF%85%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Ευκλείδεια γεωμετρία">ευκλείδεια γεωμετρία</a></li> <li><a href="/wiki/%CE%A3%CF%86%CE%B1%CE%B9%CF%81%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Σφαιρική γεωμετρία">σφαιρική γεωμετρία</a></li> <li><a href="/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B2%CE%BF%CE%BB%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Υπερβολική 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href="/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Παραστατική γεωμετρία">παραστατική γεωμετρία</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">υποπεδία της γεωμετρίας</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/w/index.php?title=%CE%95%CF%80%CE%B9%CF%80%CE%B5%CE%B4%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1&action=edit&redlink=1" class="new" title="Επιπεδομετρία (δεν έχει γραφτεί ακόμα)">επιπεδομετρία</a></li> <li><a href="/wiki/%CE%A3%CF%84%CE%B5%CF%81%CE%B5%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Στερεομετρία">στερεομετρία</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10387572"><link rel="mw-deduplicated-inline-style" 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</div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ανακτήθηκε από "<a dir="ltr" href="https://el.wikipedia.org/w/index.php?title=Τριγωνομετρία&oldid=10841562">https://el.wikipedia.org/w/index.php?title=Τριγωνομετρία&oldid=10841562</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%CE%95%CE%B9%CE%B4%CE%B9%CE%BA%CF%8C:%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B5%CF%82" title="Ειδικό:Κατηγορίες">Κατηγορία</a>: <ul><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%BF%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Κατηγορία:Τριγωνομετρία">Τριγωνομετρία</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Κρυμμένες κατηγορίες: <ul><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1_Commons_%CE%BC%CE%B5_%CF%84%CE%AF%CF%84%CE%BB%CE%BF_%CF%83%CE%B5%CE%BB%CE%AF%CE%B4%CE%B1%CF%82_%CE%B4%CE%B9%CE%B1%CF%86%CE%BF%CF%81%CE%B5%CF%84%CE%B9%CE%BA%CF%8C_%CE%B1%CF%80%CF%8C_%CF%84%CF%89%CE%BD_Wikidata" title="Κατηγορία:Κατηγορία Commons με τίτλο σελίδας διαφορετικό από των Wikidata">Κατηγορία Commons με τίτλο σελίδας διαφορετικό από των Wikidata</a></li><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:Commonscat_%CF%80%CE%BF%CF%85_%CF%84%CF%81%CE%B1%CE%B2%CE%AC%CE%B5%CE%B9_%CE%B4%CE%B5%CE%B4%CE%BF%CE%BC%CE%AD%CE%BD%CE%B1_%CE%B1%CF%80%CF%8C_Wikidata" title="Κατηγορία:Commonscat που τραβάει δεδομένα από Wikidata">Commonscat που τραβάει δεδομένα από Wikidata</a></li><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:%CE%A3%CF%8D%CE%BD%CE%B4%CE%B5%CF%83%CE%BC%CE%BF%CE%B9_wayback_%CF%80%CF%81%CE%BF%CF%84%CF%8D%CF%80%CE%BF%CF%85_Webarchive" title="Κατηγορία:Σύνδεσμοι wayback προτύπου Webarchive">Σύνδεσμοι wayback προτύπου Webarchive</a></li><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:%CE%9B%CE%AE%CE%BC%CE%BC%CE%B1%CF%84%CE%B1_%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1%CF%82_%CE%BC%CE%B5_%CE%B1%CE%BD%CE%B1%CE%B3%CE%BD%CF%89%CF%81%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AC_LCCN" title="Κατηγορία:Λήμματα Βικιπαίδειας με αναγνωριστικά LCCN">Λήμματα Βικιπαίδειας με αναγνωριστικά LCCN</a></li><li><a href="/wiki/%CE%9A%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%AF%CE%B1:%CE%9B%CE%AE%CE%BC%CE%BC%CE%B1%CF%84%CE%B1_%CE%92%CE%B9%CE%BA%CE%B9%CF%80%CE%B1%CE%AF%CE%B4%CE%B5%CE%B9%CE%B1%CF%82_%CE%BC%CE%B5_%CE%B1%CE%BD%CE%B1%CE%B3%CE%BD%CF%89%CF%81%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AC_BNF" title="Κατηγορία:Λήμματα Βικιπαίδειας με αναγνωριστικά BNF">Λήμματα Βικιπαίδειας με αναγνωριστικά BNF</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Τελευταία τροποποίηση 11:58, 22 Νοεμβρίου 2024.</li> <li id="footer-info-copyright">Όλα τα κείμενα είναι διαθέσιμα υπό την <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.el">Creative Commons Attribution-ShareAlike License</a>· μπορεί να ισχύουν και πρόσθετοι όροι. 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