SECTION 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 443 OBJECTIVES 1 Graph the Tangent Function = y x tan and the Cotangent Function = y x cot (p. 443) 2 Graph Functions of the Form ω( ) = + y A x B tan and ω( ) = + y A x B cot (p. 445) 3 Graph the Cosecant Function = y x csc and the Secant Function = y x sec (p. 447) 4 Graph Functions of the Form ω( ) = + y A x B csc and ω( ) = + y A x B sec (p. 448) 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions Now Work the ‘Are You Prepared?’ problems on page 448. • Vertical Asymptotes (Section 4.5, pp. 240–241) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Graph the Tangent Function tan = y x and the Cotangent Function cot = y x Because the tangent function has period π, we only need to determine the graph over some interval of length π. The rest of the graph consists of repetitions of that graph. Because the tangent function is not defined at π π π π − − , 3 2 , 2 , 2 , 3 2 , , we concentrate on the interval π π ( ) − 2 , 2 , of length π, and construct Table 8, which lists some points on the graph of = y x tan , π π − < < x 2 2 . We plot the points in the table and connect them with a smooth curve. See Figure 66 for a partial graph of = y x tan , where π π − ≤ ≤ x 3 3 . Figure 66 π π = − ≤ ≤ y x x tan , 3 3 x y 3 1 21 3 2 2 3 2 2 3 ( , 1) (0, 0) ( , ) ( , ) ( , ) 2( , 21) 2( , 2 ) p –– 2 2 p –– 3 2 p –– 6 2 p –– 6 p –– 6 p –– 6 p –– 2 p –– 4 p –– 4 p –– 3 p –– 3 p –– 3 3 –– 3 3 –– 3 3 –– 3 3 –– 3 x = y x tan x y, ( ) π − 3 − ≈ − 3 1.73 π ( ) − − 3 , 3 π − 4 −1 π ( ) − − 4 , 1 π − 6 − ≈ − 3 3 0.58 π − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 6 , 3 3 0 0 ( ) 0, 0 π 6 ≈ 3 3 0.58 π⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 6 , 3 3 π 4 1 π( ) 4 , 1 π 3 ≈ 3 1.73 π( ) 3 , 3 Table 8 To complete one period of the graph of = y x tan , investigate the behavior of the function as x approaches π − 2 and π 2 . Be careful, though, because = y x tan is not defined at these numbers. To determine this behavior, use the identity = x x x tan sin cos
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