Trigonometric Functions

(Please review Trigonometry under Algebra/Precalculus Review on the class webpage.)
In this section we will look at the derivatives of the trigonometric functions
sin x; cos x; tan x ; sec x; csc x; cot x:
Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are dened and
continuous everywhere and
tan x =
sin x
cos x
; sec x =
1
cos x
; csc x =
1
sin x
; cot x =
cos x
sin x
;
are continuous on their domains (all values of x where the denominator is non-zero). The graphs of the
above functions are shown at the end of this lecture to help refresh your memory: Before we calculate
the derivatives of these functions, we will calculate two very important limits.
First Important Limit
lim
!0
sin 

= 1:
See the end of this lecture for a geometric proof of the inequality,
sin  <  < tan :
shown in the picture below for  > 0,
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
– 0.2
– 0.4
– 0.6
– 0.8
– 1
– 1.2
– 1.4
– 1.6
– 1 – 0.5 0.5 1 1.5
|AD| = tan !
!
E
D
C A
B
O
sin !
!
1
From this we can easily derive that
cos  <
sin 

< 1
and we can use the squeeze theorem to prove that the limit shown above is 1.
1
Another Important Limit
From the above limit, we can derive that :
lim
!0
cos  ?? 1

= 0
Example Calculate the limits:
lim
x!0
sin 5x
sin 3x
; lim
x!0
sin(x3)
x
:
Derivatives of Trigonometric Functions
1. From our trigonometric identities, we can show that
d
dx
sin x = cos x :
d
dx
sin x = lim
h!0
sin(x + h) ?? sin(x)
h
= lim
h