7.7 *Elementary Transcendental Functions 431Apply Theorems 6.1.13 and 6.1.14. •Corollary 7. 7. 27 The sine function is differentiable on ( - i,^3 ;), and
-Slnd. X = { Vl-sin 2' 2.
(^2) x ifx E (-"- "-]}
dx -Vl - sin^2 x if x E [i,^3 ;)
Corollary 7. 7 .28 The sine function is differentiable everywhere, and
d. { Vl - sin^2 x if x - 2mr E [-i, i] for some integer n}
-SlnX =.
dx -)l - sin^2 x if x - 2nn E [i,^3 ;] for some integer n
Proof. Prove differentiability at -i and apply periodicity (see Exercise
6.2.19) .•
THE COSINE FUNCTION^21
Definition 7.7.29 (The Cosine Function) (a) On [-i,^3 ;], define cosx
by
cosx= 2'2.
{
Vl - sin^2 x if x E [-"- "-] }
-Vl - sin^2 x if x E [i,^3 ;]
(b) Note that cos ( - i) = cos (3; ) = 0, and cos 0 = l.
(c) To extend the cosine function periodically, with period 2n, to (-oo, +oo ),
we appeal to Exercise 6.2.19.
(d) Note that cosine is an even function. [See Exercise 6.2.5.]
(e) Note that Vx E IR, sin^2 x + cos^2 x = l.
DERIVATIVES OF SINE AND COSINE
Theorem 7. 7 .30 The sine and cosine functions are differentiable everywhere,
d. d.
andVxEIR, dxsmx=cosx and dxcosx=-smx.
Proof. First prove these results true on [ -i,^3 ;] , and then use the chain
rule and periodicity (see Exercise 6.2.20) to extend them to all of ( -oo, +oo).
Finally, we come to the matter of the trigonometric identities. The following
theorem establishes the crucial ones.
- For a definition based on infinite series, see Section 8.8.