Trigonometric identities are equations that relate to different trigonometric functions and are true for any value of the variable that is there in the\u00a0domain. Basically, an identity is an equation that holds true for all the values of the variable(s) present in it.<\/p>\n\n\n\n
For example, some of the\u00a0algebraic identities\u00a0are: The algebraic identities relate just the variables whereas the trig identities relate the 6 trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. Let’s learn about each type of trigonometric identities in detail.<\/p>\n\n\n\n We already know that the reciprocals of sin, cosine, and tangent are cosecant, secant, and cotangent respectively.<\/p>\n\n\n\n Thus, the reciprocal identities are given as,<\/p>\n\n\n\n The Pythagorean trigonometric identities in trigonometry are derived from the\u00a0Pythagoras theorem. Applying Pythagoras theorem to the right-angled triangle below, we get:<\/p>\n\n\n\n Opposite2 <\/sup>+ Adjacent2 <\/sup>= Hypotenuse2<\/sup><\/p>\n\n\n\n Dividing both sides by Hypotenuse2<\/sup><\/p>\n\n\n\n Opposite2<\/sup>\/Hypotenuse2<\/sup> + Adjacent2<\/sup>\/Hypotenuse2<\/sup> = Hypotenuse2<\/sup>\/Hypotenuse2<\/sup><\/p>\n\n\n\n This is one of the Pythagorean identities. In the same way, we can derive two other Pythagorean trigonometric identities.<\/p>\n\n\n\n The complementary angles are a pair of two angles such that their sum is equal to 90\u00b0. The\u00a0complement\u00a0of an angle \u03b8 is (90 – \u03b8). The trigonometric ratios of complementary angles are:<\/p>\n\n\n\n The supplementary angles are a pair of two angles such that their sum is equal to 180\u00b0. The\u00a0supplement\u00a0of an angle \u03b8 is (180 – \u03b8). The trigonometric ratios of supplementary angles are:<\/p>\n\n\n\n The sum and difference identities include the formulas of sin(A+B), cos(A-B), cot(A+B), etc.<\/p>\n\n\n\n Double angle formulas: <\/strong>The double angle trigonometric identities can be obtained by using the sum and difference formulas.<\/p>\n\n\n\n For example, from the above formulas:<\/p>\n\n\n\n sin (A+B) = sin A cos B + cos A sin B<\/p>\n\n\n\n Substitute A = B = \u03b8 on both sides here, we get:<\/p>\n\n\n\n sin (\u03b8 + \u03b8) = sin\u03b8 cos\u03b8 + cos\u03b8 sin\u03b8 In the same way, we can derive the other double-angle identities.<\/p>\n\n\n\n Using one of the above double angle formulas, Replacing \u03b8 by \u03b8\/2 on both sides,<\/p>\n\n\n\n sin (\u03b8\/2) = \u00b1\u221a[(1 – cos \u03b8)\/2]<\/p>\n\n\n\n This is the half-angle formula of sin.<\/p>\n\n\n\n In the same way, we can derive the other half-angle formulas.<\/p>\n\n\n\n sin (\u03b8\/2) = \u00b1\u221a[(1 – cos\u03b8)\/2]<\/p>\n\n\n\n cos (\u03b8\/2) = \u00b1\u221a(1 + cos\u03b8)\/2<\/p>\n\n\n\n tan (\u03b8\/2) = \u00b1\u221a[(1 – cos\u03b8)(1 + cos\u03b8)]<\/p>\n\n\n\n The trigonometric identities that we have learned are derived using the right-angled triangles. There are a few other identities that we use in the case of triangles that are not right-angled.<\/p>\n","protected":false},"excerpt":{"rendered":" Trigonometric identities are equations that relate to different trigonometric functions and are true for any value of the variable that is there in the\u00a0domain. Basically, an identity is an equation that holds true for all the values of the variable(s) present in it. For example, some of the\u00a0algebraic identities\u00a0are:(a + b)2\u00a0= a2\u00a0+ 2ab + b2(a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[310],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7751"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=7751"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7751\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=7751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=7751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=7751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
(a + b)2\u00a0<\/sup>= a2\u00a0<\/sup>+ 2ab + b2<\/sup>
(a – b)2\u00a0<\/sup>= a2\u00a0<\/sup>– 2ab+ b2<\/sup>
(a + b)(a-b)= a2\u00a0<\/sup>– b2<\/sup><\/p>\n\n\n\nReciprocal Trigonometric Identities<\/strong><\/h3>\n\n\n\n
Pythagorean Trigonometric Identities<\/strong><\/h3>\n\n\n\n
Complementary and Supplementary Trigonometric Identities<\/h2>\n\n\n\n
Sum and Difference Trigonometric Identities<\/h2>\n\n\n\n
Double and Half Angles Trigonometric Identities<\/h2>\n\n\n\n
sin 2\u03b8 = 2 sin\u03b8 cos\u03b8<\/p>\n\n\n\n
= 2 cos2\u03b8 – 1
= 1 – sin 2 \u03b8<\/li>Half Angle Formulas<\/strong><\/h3>\n\n\n\n
cos 2\u03b8 = 1 – 2 sin2<\/sup>\u03b8
2 sin2<\/sup>\u03b8 = 1- cos 2\u03b8
sin2<\/sup>\u03b8 = (1 – cos2\u03b8)\/(2)
sin \u03b8 = \u00b1\u221a[(1 – cos 2\u03b8)\/2]<\/p>\n\n\n\n