{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

M1410407Part1

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

This preview shows pages 1–4. Sign up to view the full content.

(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 .

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

View Full Document
(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.
(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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}