416 CHAPTER 6 Trigonometric Functions Finding Exact Values Using Periodic Properties Find the exact value of each of the following angles: (a) π sin 17 4 (b) π ( ) cos 5 (c) π tan 5 4 Solution EXAMPLE 1 (a) It is best to sketch the angle first, as shown in Figure 41(a). Since the period of the sine function is π2 , each full revolution can be ignored leaving the angle π 4 . Then π π π π ( ) = + = = sin 17 4 sin 4 4 sin 4 2 2 (b) See Figure 41(b). Since the period of the cosine function is π2 , each full revolution can be ignored leaving the angle π. Then π π π π ( ) ( ) = + = = − cos 5 cos 4 cos 1 (c) See Figure 41(c). Since the period of the tangent function is π, each halfrevolution can be ignored leaving the angle π 4 . Then π π π π ( ) = + = = tan 5 4 tan 4 tan 4 1 y x u 5 5p ––– 4 (c) Figure 41 y x u 5 17p ––– 4 (a) y x u 5 5p (b) The periodic properties of the trigonometric functions will be very helpful to us when we study their graphs later in the chapter. Now Work PROBLEM 11 Quadrant of P θ sin , θ csc θ cos , θ sec θ tan , θ cot I Positive Positive Positive II Positive Negative Negative III Negative Negative Positive IV Negative Positive Negative Table 5 3 Determine the Signs of the Trigonometric Functions in a Given Quadrant Let ( ) = P x y , be the point on the unit circle that corresponds to the angle θ. If we know in which quadrant the point P lies, then we can determine the signs of the trigonometric functions of θ. For example, if ( ) = P x y , lies in quadrant IV, as shown in Figure 42, then we know that > x 0 and < y 0. Consequently, θ θ θ θ θ θ = < = > = < = < = > = < y x y x y x x y sin 0 cos 0 tan 0 csc 1 0 sec 1 0 cot 0 Table 5 lists the signs of the six trigonometric functions for each quadrant. Figure 43 on the next page provides two illustrations. Figure 42 θ in quadrant IV, > < x y 0, 0 y x 1 21 21 1 P 5 (x, y ), x . 0, y , 0 u
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