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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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,...
View Full Document

Page1 / 18

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online