1549901369-Elements_of_Real_Analysis__Denlinger_

430 Chapter 7 • The Riemann Integral

Definition 7.7.23 (of sinx, for All Real Numbers)

(a) For~ :::; x:::;^3 ;, we define sin x = - sin(x - 7r).
Note that when i ::::; x::::;^3 ;, - i ::::; x - 7r ::::; i' and that when x = i' both
sides of the defining equation are the same.

(b) The function sin: [ -i,^3 ;] --> IR is continuous on [ -i,^3 ; ] and sin ( -i) =
sin (3;) = -1. Thus, by Exercise 6 .2. 19 , we can extend sinx to a function that

is continuous and periodic on IR, with period^3 ; - ( -i) = 27r.

(c) Show that sinx is an odd function on R [First show it is odd on [-7r, ?r],

then show it is odd on R]

DIFFERENTIABILITY OF THE SINE FUNCTION

Lemma 7.7.24 The sine function is differentiable on (-i,i), and \:/x E

(.!!: 2'2'dx .!!:) ..!!: sinx = · v^1 1 - sin^2 x ·

dy

Proof. The function y = arcsin x is differentiable on ( -1, 1), and

dx

1

v'f=X2. Thus, by the inverse function t heorem for differentiable functions

(6 .2.4), the function x = siny is differentiable at every y E (-i, i) and

dx d. 1

• = -siny=
  dy dy^1

= v'l -x^2 = \/1 -sin^2 y. •

Lemma 7.7.25 The sine function is differentiable on (i,^3 ;), and\:/x E (i,^3 ;),
d
dx sin x = -Vl -sin^2 x.

Proof. Apply Definition 7.7.22 and the chain rule on (i,^3 ;). •

Lemma 7.7.2 6 The sine function is differentiable at i' and

when x = i, d~ sin x = 0 = J 1 - sin^2 i.

P roo f. 1 Im. -d sm. x = 1. Im v I 1 -sin^2 x

X-+11" /2-dX X-+11" /2-

= 0, by continuity of sin x on IR

= lim (-J l -sin^2 x)

X-+7r/2+

1

. d.
Im -d sm x.
X-+7r/2+ X