418 CHAPTER 6 Trigonometric Functions Finding Exact Values Using Identities When Sine and Cosine Are Given Given θ = sin 5 5 and θ = cos 2 5 5 , find the exact values of the four remaining trigonometric functions of θ using identities. Solution EXAMPLE 3 Based on a quotient identity from (3), we have θ θ θ = = = tan sin cos 5 5 2 5 5 1 2 Then we use the reciprocal identities from (2) to get θ θ θ θ θ θ = = = = = = = = = = = csc 1 sin 1 5 5 5 5 5 sec 1 cos 1 2 5 5 5 2 5 5 2 cot 1 tan 1 1 2 2 Now Work PROBLEM 3 5 The equation of the unit circle is + = x y 1 2 2 or, equivalently, + = y x 1 2 2 If ( ) = P x y , is the point on the unit circle that corresponds to the angle θ, then θ = y sin and θ = x cos , so we have θ θ ( ) ( ) + = sin cos 1 2 2 (4) It is customary to write θ sin2 instead of θ θ ( ) sin , cos 2 2 instead of θ ( ) cos , 2 and so on. With this notation, we can rewrite identity (4) as θ θ + = sin cos 1 2 2 (5) θ θ + = tan 1 sec 2 2 (6) θ θ + = cot 1 csc 2 2 (7) If θ ≠ cos 0, we can divide each side of identity (5) by θ cos . 2 sin cos cos cos 1 cos sin cos 1 1 cos 2 2 2 2 2 2 2 θ θ θ θ θ θ θ θ ( ) ( ) + = + = Now use identities (2) and (3) to get Similarly, if θ ≠ sin 0, we can divide both sides of identity (5) by θ sin2 and use identities (2) and (3) to get θ θ + = 1 cot csc , 2 2 which we write as
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