444 CHAPTER 6 Trigonometric Functions See Table 9. If x is close to π ≈ 2 1.5708 but remains less than π 2 , then x sin is close to 1, and x cos is positive and close to 0. (To see this, refer to the graphs of the sine function and the cosine function.) The ratio x x sin cos is positive and large, so, as π →x 2 from the left, then →∞ x tan . Using limit notation, = ∞ π → − x lim tan . x 2 In other words, the vertical line π = x 2 is a vertical asymptote of the graph of = y x tan . x x sin cosx = tan y x π ≈ 3 1.05 3 2 1 2 ≈ 3 1.73 1.5 0.9975 0.0707 14.1 1.57 0.9999 × − 7.96 10 4 1255.8 1.5707 0.9999 × − 9.6 10 5 10,381 π ≈ 2 1.5708 1 0 Undefined Table 9 Properties of the Tangent Function • The domain is the set of all real numbers, except odd multiples of π 2 . • The range is the set of all real numbers. • The tangent function is an odd function, as the symmetry of the graph with respect to the origin indicates. • The tangent function is periodic, with period π. • The x -intercepts are π π π π π − − … … , 2 , ,0, ,2 ,3 , ; the y -intercept is 0. • Vertical asymptotes occur at π π π π = − − … … x , 3 2 , 2 , 2 , 3 2 , If x is close to π − 2 but remains greater than π − 2 , then sin x is close to −1, cos x is positive and close to 0, and the ratio x x sin cos is negative and large. That is, as π →− x 2 from the right, then →−∞ x tan . Using limit notation, = −∞ π →− + x lim tan . x 2 In other words, the vertical line π = − x 2 is also a vertical asymptote of the graph of = y x tan . With these observations, one period of the graph can be completed. Obtain the complete graph of = y x tan by repeating this period, as shown in Figure 67. Figure 68 shows the graph using a TI-84 Plus CE. Figure 67 = −∞< <∞ y x x x tan , , not equal to odd multiples of π −∞< <∞ y 2 , x y 1 21 22p 2p 2p p p–– 2 p–– 2 2 2 5p ––– 2 3p ––– 2 5p ––– 2 2 3p ––– 2 x 5 x 5 x 5 x 5 x 5 x 5 (2 , 21) p–– 4 ( , 1) p–– 4 Figure 68 6 26 22p 2p The graph of = y x tan in Figure 67 illustrates the following properties. Now Work PROBLEMS 7 AND 15
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