1549901369-Elements_of_Real_Analysis__Denlinger_

7.7 *Elementary Transcendental Functions 429

( c) arcsin x is differentiable on ( -1, 1), and dd = ~.

x l - x^2

(d) arcsinx is an odd^19 function on (-l, 1).

Remarks 7.7.20 (a) arcsin(-1,1) is an interval. [See Theorem 5.3.8. J

(b) arcsinx is bounded on (-1,1). [Show that jarcsinxl < 2.]

(c) Since arcsinx is continuous, strictly increasing, and bounded on (-1, 1),

Corollary 5.3.14 assures us that we can extend arcsin x to a continuous, strictly

increasing function f on the closed interval [-1, 1 J, and

arcsin[-1, l] = [c,d]

where c = inf{arcsinx: -1 < x < l} = lim arcsin x = lim r h
x-+-1+ x-+-1+ lo 1 - t2

and d = sup{arcsinx: -1 < x < l } = lim arcsin x = lim r h·
x-+1- x-+1-lo l - t^2

Definition 7. 7 .21 (Definition of 7r)

7r = 2 arcsin 1 = 2 sup{ arcsin x : -1 < x < 1}

= 2 lim arcsin x

= 2 lim.

1

x dt

X-+ 1 - 0 v'1-=t2

THE SINE FUNCTION^20

Definition 7.7.22 (of sinx, for -~ :s; x :s; ~)

Since arcsin : [-1, 1 J __, [-~, ~], and is 1-1 and onto, it has an inverse func-
tion, which we call the sine function. Thus,

sin : [ -~ , ~ J __, [ -1, 1 J.

Note that:
(a) By Definition 7.7.21, ~ = arcsin l ; that is, sin~= l.
(b) By Corollary 5.5.3, sin xis continuous and strictly increasing on [-~, ~).

( c) sin x is an odd function on [-~, ~). In fact, the inverse of any invertible
odd function is odd.

19. For definitions of even and odd functions, see Exercise 6.2.5.


20. For a definition based on infinite series, see Section 8.8.