SECTION 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 447 Note in Figure 72(c) that the period of ( ) = y x 3cot2 is π 2 because of the compression of the original period π by a factor of 1 2 . Notice that the asymptotes are π π π π = − = = = = x x x x x 2 , 0, 2 , , 3 2 , and so on, also because of the compression. Now Work PROBLEM 23 3 Graph the Cosecant Function csc = y x and the Secant Function sec = y x The cosecant and secant functions, sometimes referred to as reciprocal functions, are graphed by making use of the reciprocal identities = = x x x x csc 1 sin and sec 1 cos For example, the value of the cosecant function = y x csc at a given number x equals the reciprocal of the corresponding value of the sine function, provided that the value of the sine function is not 0. If the value of sin x is 0, then x is an integer multiple of π. At those numbers, the cosecant function is not defined. In fact, the graph of the cosecant function has a vertical asymptote at each integer multiple of π. Figure 73 shows the graph. Figure 73 = −∞< <∞ y x x csc , , x not equal to integer multiples of π ≥ y , 1 x y 1 ( , 1) 52 2} 2} } } 22 2 2 22 2 2 2 52 5 5 5 5 5 Using the idea of reciprocals, the graph of = y x sec is obtained in a similar manner. See Figure 74. 4 24 22p 2p Figure 74 = −∞< <∞ y x x x sec , , not equal to odd multiples of π ≥ y 2 , 1 3p ––– 2 x 5 2 3p––– 2 x 5 p–– 2 x 5 2 p–– 2 x 5 y 5 sec x x y 1 (2p, 21) (p, 21) (0, 1) y 5 cos x 2p 2p p 21
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