422 CHAPTER 6 Trigonometric Functions In Words Cosine and secant are even functions; the others are odd functions. Now we know θ = tan 1 2 and θ = − sec 5 2 . Using reciprocal identities, we find cos 1 sec 1 5 2 2 5 2 5 5 cot 1 tan 1 1 2 2 θ θ θ θ = = − = − = − = = = To find θ sin , use the following reasoning: tan sin cos so sin tan cos 1 2 2 5 5 5 5 csc 1 sin 1 5 5 5 5 5 θ θ θ θ θ θ θ θ ( ) ( )( ) = = = ⋅ − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = = − = − = − Now Work PROBLEM 4 3 6 Use Even–Odd Properties to Find the Exact Values of the Trigonometric Functions Recall that a function f is even if θ θ ( ) ( ) − = f f for all θ in the domain of f; a function f is odd if θ θ ( ) ( ) − = − f f for all θ in the domain of f. We will now show that the trigonometric functions sine, tangent, cotangent, and cosecant are odd functions and the functions cosine and secant are even functions. THEOREM Even–Odd Properties θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) − = − − = − = − − = − − = − = − sin sin cos cos tan tan csc csc sec sec cot cot Proof Let ( ) = P x y , be the point on the unit circle that corresponds to the angle θ. See Figure 46. Using symmetry, the point Q on the unit circle that corresponds to the angle θ− will have coordinates ( ) − x y , . Using the definition of the trigonometric functions, we have θ θ θ θ ( ) ( ) = − = − = − = y y x x sin sin cos cos so θ θ θ θ ( ) ( ) − = − = − − = = y x sin sin cos cos Now, using these results and some of the fundamental identities, we have Figure 46 y x u 2u A 5 (1, 0) 21 1 21 1 Q 5 (x, 2y) P 5 (x, y) O θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − = − − = − = − − = − = − = − − = − = = − = − = − = − tan sin cos sin cos tan cot 1 tan 1 tan cot sec 1 cos 1 cos sec csc 1 sin 1 sin csc ■
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