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4265 TRIGONOMETRIC FUNCTIONSSection 5-9 Inverse Trigonometric FunctionsInverse Sine FunctionInverse Cosine FunctionInverse Tangent FunctionSummaryInverse Cotangent, Secant, and Cosecant Functions (Optional)A brief review of the general concept of inverse functions discussed in Section4-2 should prove helpful before proceeding with this section. In the following boxwe restate a few important facts about inverse functions from that section.FACTS ABOUT INVERSE FUNCTIONSFor fa one-to-one function and f21its inverse:1.If (a,b) is an element of f, then (b,a) is an element of f21, andconversely.2.Range of f5Domain of f21Domain of f5Range of f213.4.If x5f21(y), then y5f(x) for yin the domain of f21and xin thedomain of f, and conversely.5.f(f21(y)) 5yfor yin the domain of f21f21(f(x)) 5xfor xin the domain of fAll trigonometric functions are periodic; hence, each range value can be asso-ciated with infinitely many domain values (Fig. 1). As a result, no trigonometricfunction is one-to-one. Without restrictions, no trigonometric function has aninverse function. To resolve this problem, we restrict the domain of each func-tion so that it is one-to-one over the restricted domain. Thus, for this restricteddomain, an inverse function is guaranteed.xyy5f(x)x5f21(y)ff21fDOMAIN fRANGE ff21(y)xRANGE f21DOMAIN f21yf(x)
5-9 Inverse Trigonometric Functions427Inverse trigonometric functions represent another group of basic functions thatare added to our library of elementary functions. These functions are used in manyapplications and mathematical developments, and will be particularly useful to uswhen we solve trigonometric equations in Section 6-5.Inverse Sine FunctionHow can the domain of the sine function be restricted so that it is one-to-one?This can be done in infinitely many ways. A fairly natural and generally acceptedway is illustrated in Figure 2.If the domain of the sine function is restricted to the interval [2p/2, p/2], wesee that the restricted function passes the horizontal line test (Section 4-2) andthus is one-to-one. Note that each range value from 21 to 1 is assumed exactlyonce as xmoves from 2p/2 to p/2. We use this restricted sine function to definethe inverse sine function.INVERSE SINE FUNCTIONThe inverse sine function,denoted by sin21or arcsin, is defined as theinverse of the restricted sine function y5sin x, 2p/2 #x#p/2. Thus,y5sin21xand y5arcsin xare equivalent tosin y5xwhere2p/2 #y#p/2, 21 #x#1In words, the inverse sine of x, or the arcsine of x, is the number orangle y, 2p/2 #y#p/2, whose sine is x.To graph y5sin21x, take each point on the graph of the restricted sine func-tion and reverse the order of the coordinates. For example, since (2p/2, 21), xyp2p221021FIGURE 2y5sin xis one-to-one over[2p/2, p/2].xy24p22p2p014pFIGURE 1y5sin xis not one-to-one over(2`, `).DEFINITION1