1. Trigonometric functions
3.4 Derivatives of the
Trigonometric functions
Rita Korsunsky

2. 1 - cos x 0 < sin x < x 0 < < x P (cos x , sin x)x 1
P (cos x , sin x)
In the Unit circle (Radius=1):
P(cos x, sin x)
sin x
1-cos x
cos x
0 <
sin x
< x
1 - cos x
0 <
< x
By Sandwich Theorem :

3. Let x be any number between 0 and .
Proof
Let x be any number between 0 and .
Area ∆OMP = OM • MP
= cos x sin x
Area ∆OUQ = • 1 • UQ = C u v O M x P
cos x
sin x
U (1, 0)
tan x Q

4. Trigonometric Limits

5. Example 1

6. Derivatives of Trigonometric Functions

7. PROOF

8. PROOF

9. PROOF

10. PROOF

11. Example 2
Solution By quotient rule

12. Example 3.
Find g'(x) if g(x) = sec x tan x
Solution: By product rule

13. Example 4.
Find the slopes of the tangent lines to the graph of y=sin x, at the points with x coordinates
(b) For what values of x is the tangent line horizontal?
Solution
The slope of the tangent line at (x,y) is
(b) A tangent line is horizontal if its slope is zero.
y ' = 0; that is, cos x = 0

14. Example 5
Find an equation of the normal line to the graph of y = tan x at point P(/4, 1)
Solution
slope of the normal line is
Equation of the normal line is: or

15. Example 6 Find the first eight derivatives of f (x) = sin x Solution
Since f (x) = sin x, it follows that if we continue differentiating, the same pattern repeats; that is