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M1410407Part1 - (Section 4.7 Inverse Trig Functions 4.72...

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(Section 4.7: Inverse Trig Functions) 4.72 SECTION 4.7: INVERSE TRIG FUNCTIONS You may want to review Section 1.8 on inverse functions. PART A : GRAPH OF sin - 1 x (or arcsin x ) Warning: Remember that f 1 denotes function inverse, not multiplicative inverse (or reciprocal). Usually, f 1 1 f . In particular, sin 1 x 1 sin x , or csc x . We can say that sin x ( ) 1 = 1 sin x = csc x . Although it is often helpful in Calculus to rewrite sin n x as sin x ( ) n , this is not true of sin 1 x , because 1 is not an exponent in that case. However, 1 does act as an exponent in sin x ( ) 1 . If f x ( ) = sin x , and the domain is R (which is, after all, the implied domain), then f is not a one-to-one function, and it has no inverse function . We want to define an inverse sine (or “arcsine”) function f 1 x ( ) = sin - 1 x (or arcsin x ) . To do so, we must restrict the domain of f x ( ) = sin x so that it is a one-to-one function whose graph passes the HLT (Horizontal Line Test). What should this restricted domain be? It should be an x -interval on which the sin x graph: 1) Passes the HLT, and 2) Is as “tall” as the original, unrestricted sin x graph. In other words, we would like the range to be the same as before. It is universally agreed that we take the x -interval π 2 , π 2 as our restricted domain. The resulting range for our sin x function remains 1,1 .
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(Section 4.7: Inverse Trig Functions) 4.73 The resulting graph is in red below: (The x - and y -axes are scaled differently.) Observe that: • The function increases on the interval π 2 , π 2 . • The graph switches from concave up to concave down at 0, 0 ( ) . It may be easier to remember that the graph is a snake of finite length that has horizontal (one-sided) tangent lines (in green) at its endpoints.
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(Section 4.7: Inverse Trig Functions) 4.74 The graph of f 1 x ( ) = sin - 1 x (or arcsin x ) , the arcsine function, is obtained by switching the x - and y -coordinates of all the points on the red graph we just saw. (Reflecting the red
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