400 CHAPTER 6 Trigonometric Functions Solution Finding the Exact Values of the Six Trigonometric Functions of Quadrantal Angles Find the exact values of the six trigonometric functions of each of the following angles: (a) 0 0 θ = = ° (b) 2 90 θ π = = ° (c) 180 θ π = = ° (d) 3 2 270 θ π = = ° EXAMPLE 2 Figure 23 3 2 270 θ π = = ° 1 1 (d) y x P 5 (0, 21) 21 21 u 5 5 270° 2 3p Figure 23 0 0 θ = = ° 1 1 21 21 y x P 5 (1, 0) (a) u 5 0 5 0° Figure 23 (c) 180 θ π = = ° 1 1 y x P 5 (21, 0) u 5 p 5 180° 21 21 (a) The point on the unit circle that corresponds to 0 0 θ = = ° is P 1, 0 . ( ) = See Figure 23(a). Then using x 1 = and y 0 = y x y x x sin0 sin0 0 cos0 cos0 1 tan0 tan0 0 sec0 sec0 1 1 = ° = = = ° = = = ° = = = ° = = Since the y-coordinate of P is 0, csc0 and cot0 are not defined. (b) The point on the unit circle that corresponds to 2 90 θ π = = ° is P 0, 1 . ( ) = See Figure 23(b). Then y x y x y sin 2 sin90 1 cos 2 cos90 0 csc 2 csc90 1 1 cot 2 cot90 0 π π π π = ° = = = ° = = = ° = = = ° = = Since the x-coordinate of P is 0, tan 2 π and sec 2 π are not defined. (c) The point on the unit circle that corresponds to 180 θ π = = ° is P 1, 0 . ( ) = − See Figure 23(c). Then y x y x x sin sin180 0 cos cos180 1 tan tan180 0 sec sec180 1 1 π π π π = ° = = = ° = = − = ° = = = ° = = − Since the y-coordinate of P is 0, cscπ and cotπ are not defined. (d) The point on the unit circle that corresponds to 3 2 270 θ π = = ° is ( ) = − P 0, 1. See Figure 23(d). Then y x y x y sin 3 2 sin270 1 cos 3 2 cos270 0 csc 3 2 csc270 1 1 cot 3 2 cot270 0 π π π π = ° = = − = ° = = = ° = = − = ° = = Since the x-coordinate of P is 0, tan 3 2 π and sec 3 2 π are not defined. Table 2 summarizes the values of the trigonometric functions found in Example 2. Quadrantal Angles θ (Radians) θ (Degrees) sinθ cosθ tanθ cscθ secθ cotθ 0 °0 0 1 0 Not defined 1 Not defined π 2 ° 90 1 0 Not defined 1 Not defined 0 π ° 180 0 −1 0 Not defined −1 Not defined π3 2 ° 270 −1 0 Not defined −1 Not defined 0 Table 2 Figure 23 (b) 2 90 θ π = = ° 1 21 21 1 y x P 5 (0, 1) u 5 5 2 90° p
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