{"id":6546,"date":"2026-07-13T11:38:16","date_gmt":"2026-07-13T16:38:16","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=6546"},"modified":"2026-07-13T11:41:31","modified_gmt":"2026-07-13T16:41:31","slug":"angle-addition-identities","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/angle-addition-identities\/","title":{"rendered":"Angle Addition Identities"},"content":{"rendered":"\n

You can determine the value of a trigonometric function of a given angle when you can express the sum (or difference) of angles that can comprise that angle.<\/p>\n\n\n\n

\n\n \n
\n\n sin(\u03b1 + \u03b2) = sin \u03b1 cos \u03b2 + sin \u03b2 cos \u03b1\n

\n\n sin(\u03b1 \u2212 \u03b2) = sin \u03b1 cos \u03b2 \u2212 sin \u03b2 cos \u03b1\n

\n\n cos(\u03b1 + \u03b2) = cos \u03b1 cos \u03b2 \u2212 sin \u03b1 sin \u03b2\n

\n\n cos(\u03b1 \u2212 \u03b2) = cos \u03b1 cos \u03b2 + sin \u03b1 sin \u03b2\n\n <\/div>\n\n \n
\n\n tan(\u03b1 + \u03b2) =\n\n \n \n tan \u03b1 + tan \u03b2\n <\/span>\n \n 1 \u2212 tan \u03b1 tan \u03b2\n <\/span>\n <\/span>\n\n

\n\n tan(\u03b1 \u2212 \u03b2) =\n\n \n \n tan \u03b1 \u2212 tan \u03b2\n <\/span>\n \n 1 + tan \u03b1 tan \u03b2\n <\/span>\n <\/span>\n\n <\/div>\n\n<\/div>\n\n\n\n

For example, sin 75 = sin (30 + 45) = sin 30 cos 45 + sin 45 cos 30<\/p>\n\n\n\n

\n\n= \n 1<\/span>\n 2<\/span>\n<\/span>\n(\n\n \u221a2<\/span>\n 2<\/span>\n<\/span>\n)\n\n+\n\n(\n\n \u221a2<\/span>\n 2<\/span>\n<\/span>\n)\n\n(\n\n \u221a3<\/span>\n 2<\/span>\n<\/span>\n)\n\n
\n\n=\n\n\n \u221a2<\/span>\n 4<\/span>\n<\/span>\n\n+\n\n\n \u221a6<\/span>\n 4<\/span>\n<\/span>\n\n
\n\n=\n\n\n \n \u221a2 + \u221a6\n <\/span>\n 4<\/span>\n<\/span>\n\n<\/div>\n\n\n\n

Double Angle and Half Angle Identities <\/h3>\n\n\n\n

Double-angle identities are trig identities that can be used to rewrite trig functions that have a double angle.

Below are the double-angle identities and an example of how they are used.<\/p>\n\n\n\n

\n\n \n
\n\n sin(2\u03b8) = 2 sin(\u03b8) cos(\u03b8)\n\n
\n\n cos(2\u03b8) = cos\u00b2(\u03b8) \u2212 sin\u00b2(\u03b8)\n\n
\n\n = 1 \u2212 2 sin\u00b2(\u03b8)\n\n
\n\n = 2 cos\u00b2(\u03b8) \u2212 1\n\n
\n\n tan(2\u03b8) =\n\n \n \n 2 tan \u03b8\n <\/span>\n \n 1 \u2212 tan\u00b2\u03b8\n <\/span>\n <\/span>\n\n <\/div>\n\n \n
\n\n Example:<\/b> Given\n\n sin(\u03b8) =\n\n \n 4<\/span>\n 5<\/span>\n <\/span>\n\n and
\n\n 0 < \u03b8 <\n\n \n \u03c0<\/span>\n 2<\/span> \n <\/span>
\n Find sin (2\u03b8):\n\n
\n\n sin(2\u03b8) = 2\n\n (\n\n \n 4<\/span>\n 5<\/span>\n <\/span>\n\n )\n\n (\n\n \n 3<\/span>\n 5<\/span>\n <\/span>\n\n )\n\n
\n\n =\n\n \n 24<\/span>\n 25<\/span>\n <\/span>\n\n <\/div>\n\n <\/div>\n\n<\/div>\n\n\n\n

The half-angle formulas can be used in the same way.<\/p>\n\n\n\n

Half-Angle Formulas<\/h3>\n\n\n\n
\n\nsin(\u03b8\/2) = \u00b1\n\n\n \u221a<\/span>\n\n \n \n \n 1 \u2212 cos \u03b8\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n\n
\n\ncos(\u03b8\/2) = \u00b1\n\n\n \u221a<\/span>\n\n \n \n \n 1 + cos \u03b8\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n\n
\n\ntan(\u03b8\/2) = \u00b1\n\n\n \n 1 \u2212 cos \u03b8\n <\/span>\n \n sin \u03b8\n <\/span>\n<\/span>\n\n = \n\n\n \n sin \u03b8\n <\/span>\n \n 1 + cos \u03b8\n <\/span>\n<\/span>\n\n<\/div>
\n\n\n
\n
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For more help, contact The Learning Center<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

You can determine the value of a trigonometric function of a given angle when you can express the sum (or difference) of angles that can comprise that angle. sin(\u03b1 + \u03b2) = sin \u03b1 cos \u03b2 + sin \u03b2 cos \u03b1 sin(\u03b1 \u2212 \u03b2) = sin \u03b1 cos \u03b2 \u2212 sin \u03b2 cos \u03b1 cos(\u03b1 […]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-6546","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6546","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=6546"}],"version-history":[{"count":18,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6546\/revisions"}],"predecessor-version":[{"id":6566,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6546\/revisions\/6566"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6546"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}