545 5.2 Trigonometric Functions To find tan u, use the values of cos u and sin u and the quotient identity for tan u. tan u = sin u cos u = 213 4 - 23 4 = 213 4 a- 42 3b = - 213 23 = - 213 23 # 23 23 = - 239 3 Rationalize the denominator. S Now Try Exercise 145. CAUTION In exercises like Examples 9 and 10, be careful to choose the correct sign when square roots are taken. Refer as needed to the diagrams preceding Example 6 that summarize the signs of the functions. EXAMPLE 11 Using Identities to Find Function Values Find sin u and cos u, given that tan u = 4 3 and u is in quadrant III. SOLUTION Because u is in quadrant III, sin u and cos u will both be negative. It is tempting to say that since tan u = sin u cos u and tan u = 4 3 , then sin u = -4 and cos u = -3. This is incorrect, however—both sin u and cos u must be in the interval 3-1, 14. We use the Pythagorean identity tan2 u + 1 = sec2 u to find sec u, and then the reciprocal identity cos u = 1 sec u to find cos u. t an2 u + 1 = sec2 u Pythagorean identity a4 3b 2 + 1 = sec2 u tan u = 4 3 16 9 + 1 = sec2 u Square 4 3 . 25 9 = sec2 u Add. - 5 3 = sec u Choose the negative square root because sec u is negative when u is in quadrant III. - 3 5 = cos u Secant and cosine are reciprocals. Now we use this value of cos u to find sin u. sin2 u = 1 - cos2 u Pythagorean identity (alternative form) sin2 u = 1 - a- 3 5b 2 cos u = - 3 5 sin2 u = 1 - 9 25 Square - 3 5 . sin2 u = 16 25 Subtract. sin u = - 4 5 Choose the negative square root. S Now Try Exercise 143. Be careful to choose the correct sign here. Again, be careful.
RkJQdWJsaXNoZXIy NjM5ODQ=