{"id":11961,"date":"2026-03-09T14:26:00","date_gmt":"2026-03-09T05:26:00","guid":{"rendered":"https:\/\/www.manabi-edekeru.com\/?p=11961"},"modified":"2026-03-13T10:47:59","modified_gmt":"2026-03-13T01:47:59","slug":"trigonometric-identities","status":"publish","type":"post","link":"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-identities\/","title":{"rendered":"Trigonometric Identities: Clear Classification for Memorisation and Derivation"},"content":{"rendered":"\n<div class=\"wrapper\">\n<div class=\"sidebar\">\n\n\n<nav id=\"TOC\" role=\"doc-toc\">\n    <h2 id=\"toc-title\">On this page<\/h2>\n   \n  <ul>\n  <li><a href=\"#definitions-of-trigonometric-functions\" id=\"toc-definitions-of-trigonometric-functions\"><span class=\"header-section-number\">1<\/span> Definitions of Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#trigonometric-identities-proven-from-definitions\" id=\"toc-trigonometric-identities-proven-from-definitions\"><span class=\"header-section-number\">2<\/span> Trigonometric Identities Proven from Definitions<\/a><\/li>\n  <li><a href=\"#addition-formulae\" id=\"toc-addition-formulae\"><span class=\"header-section-number\">3<\/span> Addition Formulae<\/a><\/li>\n  <li><a href=\"#identities-derived-from-the-addition-formulae\" id=\"toc-identities-derived-from-the-addition-formulae\"><span class=\"header-section-number\">4<\/span> Identities Derived from the Addition Formulae<\/a><\/li>\n  <li><a href=\"#understanding-is-more-important-than-memorisation\" id=\"toc-understanding-is-more-important-than-memorisation\"><span class=\"header-section-number\">5<\/span> Understanding Is More Important Than Memorisation<\/a><\/li>\n  <\/ul>\n<\/div>\n<\/nav>\n<div class=\"main-content\">\n<p>This article summarises key trigonometric identities. We classify these identities into two groups:<\/p>\n<ol type=\"1\">\n<li>Identities proved using the definitions of trigonometric functions<br>\n<\/li>\n<li>Identities proved using the addition formulae<\/li>\n<\/ol>\n<p>This classification will help you recall the methods of proof later on.<\/p>\n<h2 data-number=\"1\" id=\"definitions-of-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">1<\/span> Definitions of Trigonometric Functions<\/h2>\n<p>The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Using a circle with radius <span class=\"math inline\">r<\/span> and a point <span class=\"math inline\">A(x, y)<\/span> on its circumference, as shown in <a href=\"#fig-class-memo-trig-formlas-trig-def\" class=\"quarto-xref\">Figure&nbsp;1<\/a>, we define the trigonometric functions as in <a href=\"#def-trigonometric-functions\" class=\"quarto-xref\">Definition&nbsp;1<\/a>.<\/p>\n<div id=\"fig-class-memo-trig-formlas-trig-def\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-class-memo-trig-formlas-trig-def-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<img decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/fig-class-memo-trig-formlas-trig-def.png\">\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-class-memo-trig-formlas-trig-def-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;1: Definitions of trigonometric functions<br>\n\n<\/figcaption>\n<\/figure>\n<\/div>\n<div id=\"def-trigonometric-functions\" class=\"theorem definition\">\n<p><span class=\"theorem-title\"><strong>Definition 1 (Definitions of Trigonometric Functions)<\/strong><\/span> <span class=\"math display\">\n\\begin{aligned}\n\\sin(\\theta) &amp;= \\frac{y}{r}\\\\\n\\cos(\\theta) &amp;= \\frac{x}{r}\\\\\n\\tan(\\theta) &amp;= \\frac{y}{x}\n\\end{aligned}\n<\/span><\/p>\n<\/div>\n<p>These definitions form the foundation of all trigonometric functions.<\/p>\n<h2 data-number=\"2\" id=\"trigonometric-identities-proven-from-definitions\" class=\"anchored\"><span class=\"header-section-number\">2<\/span> Trigonometric Identities Proven from Definitions<\/h2>\n<p>Here are identities that we can prove directly from the definitions:<\/p>\n<p><span id=\"eq-pytha-ident\"><span class=\"math display\">\n\\sin^2{\\theta} + \\cos^2{\\theta} = 1\n\\tag{1}<\/span><\/span><\/p>\n<p><span id=\"eq-sin-cos-tan\"><span class=\"math display\">\n\\tan(\\theta) = \\frac{\\sin{\\theta}}{\\cos{\\theta}}\n\\tag{2}<\/span><\/span><\/p>\n<p><span id=\"eq-tan-cos\"><span class=\"math display\">\n1 + \\tan^2{\\theta} = \\frac{1}{\\cos^2{\\theta}}\n\\tag{3}<\/span><\/span><\/p>\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/basic-trigonometric-identities\/\">Proof<\/a><\/p>\n<hr>\n<p><span id=\"eq-neg-sin\"><span class=\"math display\">\n\\sin{(-\\theta)} = -\\sin{\\theta}\n\\tag{4}<\/span><\/span><\/p>\n<p><span id=\"eq-neg-cos\"><span class=\"math display\">\n\\cos{(-\\theta)} = \\cos{\\theta}\n\\tag{5}<\/span><\/span><\/p>\n<p><span id=\"eq-neg-tan\"><span class=\"math display\">\n\\tan{(-\\theta)} = -\\tan{\\theta}\n\\tag{6}<\/span><\/span><\/p>\n<hr>\n<p><span id=\"eq-pi-neg-sin\"><span class=\"math display\">\n\\sin{(\\pi &#8211; \\theta)} = \\sin{\\theta}\n\\tag{7}<\/span><\/span><\/p>\n<p><span id=\"eq-pi-neg-cos\"><span class=\"math display\">\n\\cos{(\\pi &#8211; \\theta)} = -\\cos{\\theta}\n\\tag{8}<\/span><\/span><\/p>\n<p><span id=\"eq-pi-neg-tan\"><span class=\"math display\">\n\\tan{(\\pi &#8211; \\theta)} = -\\tan{\\theta}\n\\tag{9}<\/span><\/span><\/p>\n<hr>\n<p><span id=\"eq-pih-neg-sin\"><span class=\"math display\">\n\\sin{\\left(\\frac{\\pi}{2} &#8211; \\theta \\right)} = \\cos{\\theta}\n\\tag{10}<\/span><\/span><\/p>\n<p><span id=\"eq-pih-neg-cos\"><span class=\"math display\">\n\\cos{\\left(\\frac{\\pi}{2} &#8211; \\theta \\right)} = \\sin{\\theta}\n\\tag{11}<\/span><\/span><\/p>\n<p><span id=\"eq-pih-neg-tan\"><span class=\"math display\">\n\\tan{\\left(\\frac{\\pi}{2} &#8211; \\theta \\right)} = \\frac{1}{\\tan{\\theta}}\n\\tag{12}<\/span><\/span><\/p>\n<h2 data-number=\"3\" id=\"addition-formulae\" class=\"anchored\"><span class=\"header-section-number\">3<\/span> Addition Formulae<\/h2>\n<p>The addition formulae are essential tools that connect trigonometric functions of different angles. They are expressed as follows:<\/p>\n<div id=\"thm-addition-formulas\" class=\"theorem\">\n<p><span class=\"theorem-title\"><strong>Theorem 1 (Addition Formulae)<\/strong><\/span> &nbsp;<\/p>\n<ul>\n<li><span class=\"math inline\">\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B<\/span><br>\n<\/li>\n<li><span class=\"math inline\">\\cos(A + B) = \\cos A \\cos B &#8211; \\sin A \\sin B<\/span><br>\n<\/li>\n<li><span class=\"math inline\">\\tan(A + B) = \\frac{\\tan A + \\tan B}{1 &#8211; \\tan A \\tan B}<\/span><br>\n<\/li>\n<\/ul>\n<\/div>\n<p>You often see the addition formulae written for angles <span class=\"math inline\">A-B<\/span>. Simply substituting <span class=\"math inline\">B<\/span> with <span class=\"math inline\">-B<\/span> in <a href=\"#thm-addition-formulas\" class=\"quarto-xref\">Theorem&nbsp;1<\/a> proves this case, so we omit it here.<\/p>\n<p>Alongside the definitions, the addition formulae serve as the foundation for deriving many other identities. It is important to memorise them.<\/p>\n<h2 data-number=\"4\" id=\"identities-derived-from-the-addition-formulae\" class=\"anchored\"><span class=\"header-section-number\">4<\/span> Identities Derived from the Addition Formulae<\/h2>\n<p>Using the addition formulae, we can derive the following identities.<\/p>\n<p><strong>Double-angle identities<\/strong><\/p>\n<p><span id=\"eq-doub-ang-sin\"><span class=\"math display\">\n\\sin{2\\theta} = 2\\sin\\theta \\cos\\theta\n\\tag{13}<\/span><\/span><\/p>\n<p><span id=\"eq-doub-ang-cos\"><span class=\"math display\">\n\\cos{2\\theta} = \\cos^2\\theta &#8211; \\sin^2\\theta = 2\\cos^2\\theta &#8211; 1 = 1 &#8211; 2\\sin^2\\theta\n\\tag{14}<\/span><\/span><\/p>\n<p><span id=\"eq-doub-ang-tan\"><span class=\"math display\">\n\\tan{2\\theta} = \\frac{2\\tan\\theta}{1 &#8211; \\tan^2\\theta} \\quad (1 &#8211; \\tan^2\\theta \\neq 0)\n\\tag{15}<\/span><\/span><\/p>\n<p><strong>Half-angle identities<\/strong><\/p>\n<p><span id=\"eq-half-ang-sin\"><span class=\"math display\">\n\\sin ^{2}\\dfrac{\\alpha }{2}=\\frac{1 &#8211; \\cos \\alpha }{2}\n\\tag{16}<\/span><\/span><\/p>\n<p><span id=\"eq-half-ang-cos\"><span class=\"math display\">\n\\cos ^{2}\\dfrac{\\alpha }{2}=\\dfrac{1+\\cos \\alpha }{2}\n\\tag{17}<\/span><\/span><\/p>\n<p><span id=\"eq-half-ang-tan\"><span class=\"math display\">\n\\tan ^{2}\\dfrac{\\alpha }{2}=\\dfrac{1-\\cos \\alpha }{1+\\cos \\alpha }\n\\tag{18}<\/span><\/span><\/p>\n<p><strong>Linear Combination of sine and cosine<\/strong><\/p>\n<p><span id=\"eq-merge-trig\"><span class=\"math display\">\na\\sin \\theta + b\\cos \\theta =\\sqrt{a^{2}+b^{2}} \\sin \\left( \\theta +\\alpha \\right)\n\\tag{19}<\/span><\/span><\/p>\n<p>, where<\/p>\n<p><span class=\"math display\">\n\\begin{aligned}\n\\cos \\alpha &amp;= \\dfrac{a}{\\sqrt{a^{2}+b^{2}}},\\\\\n\\sin \\alpha &amp;= \\dfrac{b}{\\sqrt{a^{2}+b^{2}}}.\n\\end{aligned}\n<\/span><\/p>\n<h2 data-number=\"5\" id=\"understanding-is-more-important-than-memorisation\" class=\"anchored\"><span class=\"header-section-number\">5<\/span> Understanding Is More Important Than Memorisation<\/h2>\n<p>While it is convenient to memorise identities, I strongly recommend that you do not rely on rote memorisation without understanding. The proofs of trigonometric identities involve important techniques and ways of thinking in this field, which exceed mere memorisation.<\/p>\n<p>Certainly, when preparing for exams, it helps to know the identities to some extent. However, the best way to remember them naturally is to use them repeatedly. If you find you cannot remember the identities even when solving problems, you are probably not practicing enough. Rather than relying on memorisation, try to solve enough problems until the identities come to you naturally.<\/p>\n<p>In conclusion, I recommend the approach of <strong>\u201crepeatedly proving to understand the identities deeply while practising problems regularly to internalise them.\u201d<\/strong><\/p>\n<div class=\"edek-link-block\">\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/derive-trigonometric-identities-instead-of-memorising\/\">Proofs are More Important than Memorising Trigonometric Identities<\/a><\/p>\n<\/div>\n\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<ul>\n<li>A comprehensive summary of trigonometric identities<\/li>\n<li>Identities proved from their definitions <\/li>\n<li>Identities demonstrated using the addition theorems<\/li>\n<\/ul>\n","protected":false},"author":3,"featured_media":12804,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_locale":"en_US","_original_post":"https:\/\/www.manabi-edekeru.com\/?p=11961","iawp_total_views":1,"footnotes":""},"categories":[3],"tags":[],"free_or_paid":[83],"level":[55],"post_language":[49],"resource_type":[59],"topic":[82,51],"class_list":["post-11961","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","free_or_paid-free","level-intermediate","post_language-english","resource_type-reading","topic-trigonometry","topic-mathematics","en-US","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trigonometric Identities: Clear Classification for Memorisation and Derivation<\/title>\n<meta name=\"description\" content=\"This article organises 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