544 CHAPTER 5 Trigonometric Functions We give only one form of each identity. However, algebraic transformations produce equivalent forms. For example, by subtracting sin2 u from both sides of sin2 u + cos2 u = 1, we obtain an equivalent identity. cos2 U =1 −sin2 U Alternative form It is important to be able to transform these identities quickly and also to recognize their equivalent forms. Pythagorean Identities For all angles u for which the function values are defined, the following identities hold true. sin2 U +cos2 U =1 tan2 U +1 =sec2 U 1 +cot2 U =csc2 U Quotient Identities Consider the quotient of the functions sin u and cos u, for cos u ≠0. sin u cos u = y r x r = y r , x r = y r # r x = y x = tan u Similarly, cos u sin u = cot u, for sin u ≠0. Thus, we have the quotient identities. Quotient Identities For all angles u for which the denominators are not 0, the following identities hold true. sin U cos U =tan U cos U sin U =cot U EXAMPLE 10 Using Identities to Find Function Values Find sin u and tan u, given that cos u = - 23 4 and sin u 70. SOLUTION Start with the Pythagorean identity that includes cos u. sin2 u + cos2 u = 1 Pythagorean identity sin2 u + ¢- 23 4 ≤ 2 = 1 Replace cos u with - 23 4 . sin2 u + 3 16 = 1 Square - 23 4 . sin2 u = 13 16 Subtract 3 16 . sin u = { 213 4 Take square roots. sin u = 213 4 Choose the positive square root because sin u is positive. Choose the correct sign here. LOOKING AHEAD TO CALCULUS The reciprocal, Pythagorean, and quotient identities are used in calculus to find derivatives and integrals of trigonometric functions. A standard technique of integration called trigonometric substitution relies on the Pythagorean identities.
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