713 7.4 Double-Angle and Half-Angle Identities The value of tan 2u can be found in either of two ways. We can use the doubleangle identity and the fact that tan u = sin u cos u = - 4 5 3 5 = - 4 5 , 3 5 = - 4 5 # 5 3 = - 4 3 . tan 2u = 2 tan u 1 - tan2 u = 2 A - 4 3B 1 - A - 4 3B 2 = - 8 3 - 7 9 = - 8 3 # a9 7b = 24 7 Alternatively, we can find tan 2u by finding the quotient of sin 2u and cos 2u. tan 2u = sin 2u cos 2u = - 24 25 - 7 25 = - 24 25 # a25 7 b = 24 7 Same result as above S Now Try Exercise 15. EXAMPLE 2 Finding Function Values of U Given Information about 2U Find the values of the six trigonometric functions of u given cos 2u = 4 5 and 90° 6u 6180°. SOLUTION We must obtain a trigonometric function value of u alone. cos 2u = 1 - 2 sin2 u Double-angle identity 4 5 = 1 - 2 sin2 u cos 2u = 4 5 - 1 5 = -2 sin2 u Subtract 1 from each side. 1 10 = sin2 u Multiply by - 1 2 . sin u = B1 10 sin u = 12 10 # 210 210 sin u = 210 10 2a # 2a = a Now find values of cos u and tan u by sketching and labeling a right triangle in quadrant II. Because sin u = 11 10 , the triangle in Figure 6 is labeled accordingly. The Pythagorean theorem is used to find the remaining leg. cos u = -32 10 = - 3210 10 and tan u = 1 -3 = - 1 3 cos u = x r and tan u = y x We find the other three functions using reciprocals. csc u = 1 sin u = 210 , sec u = 1 cos u = - 210 3 , cot u = 1 tan u = -3 S Now Try Exercise 19. 1 –3 √10 x y 1 sin U = √10 U Figure 6 Take square roots. Choose the positive square root because u terminates in quadrant II. Quotient rule for radicals; Rationalize the denominator.
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