lec10 - 9 Derivatives of Trig Functions P K Lamm Lecture...

9. Derivatives of Trig Functions P. K. Lamm 10/02/11 (15:33) p. 1 / 5
Lecture Notes: 9. Derivatives of Trig Functions
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of definitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
The trigonometric limits
lim
x → 0
sin x
x
= 1 , and lim
x → 0
cos x - 1
x
= 0 ,
are needed in order to derive the derivative of trig functions. Recall that these limits were valid
provided that angles are measured in radians ; because of this, the derivatives found below of trig
functions also assume that angles are measured in radians.
1. Derivatives of the Sine and Cosine Functions
Both of the trig limits are needed to find the derivatives of sin x and cos x :
Theorem: (sin x )
0
= cos x .
Proof: Let f ( x ) = sin x and apply the usual definition of the derivative of f at x . That is,
f
0
( x ) = lim
h → 0
sin( x + h ) - sin x
h
.
Using the trig identity
sin( A + B ) = sin A cos B + cos A sin B,
it follows that
f
0
( x ) = lim
h → 0
sin x cos h + cos x sin h - sin x
h
= lim
h → 0
sin x
cos h - 1
h
+ cos x
sin h
h
!