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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 onetoone 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 onetoone function whose graph passes the HLT (Horizontal Line Test). What should this restricted domain be? It should be an xinterval 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 xinterval − π 2 , π 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 yaxes 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 (onesided) 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...
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.
 Fall '11
 staff
 Calculus, Inverse Functions

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