M1410407

# M1410407 - (Section 4.7: Inverse Trig Functions) 4.72...

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Unformatted text preview: (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 2 , 2 as our restricted domain. The resulting range for our sin x function remains 1,1 . (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 2 , 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. (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 graph about the line y = x may be hard to visualize.) We obtain: Observe that: The inverse function also increases, but on the interval 1,1 . The three indicated points above suggest this. However, the graph switches from concave down to concave up at 0,0 ( ) . It may be easier to remember that the graph is a snake of finite length that has vertical (one-sided) tangent lines (in green) at its endpoints. Remember that, for a pair of inverse functions, the domain of one is the range of the other. Domain Range sin x (restricted) 2 , 2 1,1 sin- 1 x (or arcsin x ) 1,1 2 , 2 (Section 4.7: Inverse Trig Functions) 4.75 PART B: GRAPH OF cos- 1 x (or arccos x ) It is universally agreed that we take the x-interval 0,...
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## M1410407 - (Section 4.7: Inverse Trig Functions) 4.72...

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