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SalasSV_07_07 - 422 CHAPTER 7 THE TRANSCENDENTAL...

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7.7 THE INVERSE TRIGONOMETRIC FUNCTIONS Since none of the trigonometric functions are one-to-one, none of them have inverses. What, then, are the inverse trigonometric functions? The Inverse Sine The graph of y = sin x is shown in Figure 7.7.1. Since every horizontal line between 1 and 1 intersects the graph at in fi nitely many points, the sine function does not have an inverse. However, observe that if we restrict the domain to the interval [ 1 2 π , 1 2 π ] (the solid portion of the graph in Figure 7.7.1), then y = sin x is one-to-one, and on that interval it takes on as a value every number in [ 1,1]. Thus, if x [ 1, 1], there is one and only one number in the interval [ 1 2 π , 1 2 π ] at which the sine function has the value x . This number is called the inverse sine of x, or the angle whose sine is x , and is written sin 1 x .
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7.7 THE INVERSE TRIGONOMETRIC FUNCTIONS 423 2 π 5 π 2 y y = sin x x y = 1 y = 1 π π 2 π 1 2 π 3 2 π 1 2 π 3 π 2 2 π 5 Figure 7.7.1 Remarks Another common notation for the inverse sine function is arcsin x , read arc sine of x . For consistency in the treatment here, we use sin 1 x throughout, but we use both notations in the Exercises. Remember that the 1 is not an exponent; do not confuse sin 1 x with the reciprocal 1 / sin x . The inverse sine function y = sin 1 x , domain: [ 1, 1], range: [ 1 2 π , 1 2 π ] is the inverse of the function Table 7.7.1 x sin x 1 2 π 1 1 3 π 1 2 3 1 4 π 1 2 2 1 6 π 1 2 0 0 1 6 π 1 2 1 4 π 1 2 2 1 3 π 1 2 3 1 2 π 1 Table 7.7.2 x sin 1 x 1 1 2 π 1 2 3 1 3 π 1 2 2 1 4 π 1 2 1 6 π 0 0 1 2 1 6 π 1 2 2 1 4 π 1 2 3 1 3 π 1 1 2 π y = sin x , domain: [ 1 2 π , 1 2 π ], range: [ 1, 1]. The graphs of these functions are shown in Figure 7.7.2. Each curve is the re fl ection of the other in the line y = x . y x y x 1 1 2 1 π 2 1 π ] y = sin x , x 2 1 π , 2 1 π [ 1 1 2 1 π 2 1 π y = sin 1 x , x [ 1, 1] Figure 7.7.2 Because these functions are inverses, (7.7.1) for all x [ 1, 1], sin (sin 1 x ) = x and (7.7.2) for all x [ 1 2 π , 1 2 π ], sin 1 (sin x ) = x . Table 7.7.1 gives some representative values of the sine function from x = − 1 2 π to x = 1 2 π . Reversing the order of the columns, we have a table for the inverse sine (Table 7.7.2).
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424 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS On the basis of Table 7.7.2 one could guess that for all x [ 1, 1], sin 1 ( x ) = − sin 1 x . This is indeed the case. Being the inverse of an odd function ( sin ( x ) = − sin x for all x [ 1 2 π , 1 2 π ] ) , the inverse sine is itself an odd function. (Verify this.) Example 1 Calculate if de fi ned: (a) sin 1 (sin 1 16 π ); (b) sin 1 (sin 7 3 π ); (c) sin (sin 1 1 3 ); (d) sin 1 (sin 9 5 π ); (e) sin (sin 1 2).
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