1. DIFFERENTIATION RULES.
3.3 Derivatives of
Trigonometric Functions
In this section, we will learn about:
Derivatives of trigonometric functions
and their applications.

2. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Let’s sketch the graph of the function f(x) = sin x and use the interpretation of f '(x) as the slope of the tangent to the sine curve in order to sketch the graph of f '.

3. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Then, it looks as if the graph of f ' may be the same as the cosine curve.

4. From the definition of a derivative, we have:
DERIVS. OF TRIG. FUNCTIONS
Equation 1
From the definition of a derivative, we have:

5. Two of these four limits are easy to evaluate.
DERIVS. OF TRIG. FUNCTIONS
Two of these four limits are easy to evaluate.

6. DERIVS. OF TRIG. FUNCTIONS
Since we regard x as a constant when computing a limit as h → 0, we have

7. The limit of (sin h)/h is not so obvious.
DERIVS. OF TRIG. FUNCTIONS
Equation 2
The limit of (sin h)/h is not so obvious.
In Example 3 in Section 2.2, we made the guess—on the basis of numerical and graphical evidence—that:

8. DERIVS. OF TRIG. FUNCTIONS

9. DERIVS. OF TRIG. FUNCTIONS
We can deduce the value of the remaining limit in Equation 1 as follows.

10. DERIVS. OF TRIG. FUNCTIONS
Equation 3

11. If we put the limits (2) and (3) in (1), we get:
DERIVS. OF TRIG. FUNCTIONS
If we put the limits (2) and (3) in (1), we get:

12. DERIV. OF SINE FUNCTION
Formula 4
So, we have proved the formula for the derivative of the sine function:

13. Differentiate y = x2 sin x.
DERIVS. OF TRIG. FUNCTIONS
Example 1
Differentiate y = x2 sin x.
Using the Product Rule and Formula 4, we have:

14. Using the same methods as in the proof of Formula 4, we can prove:
DERIV. OF COSINE FUNCTION
Formula 5
Using the same methods as in the proof of Formula 4, we can prove:

15. DERIV. OF TANGENT FUNCTION
Formula 6

16. DERIVS. OF TRIG. FUNCTIONS
We have collected all the differentiation formulas for trigonometric functions here.
Remember, they are valid only when x is measured in radians.

17. For what values of x does the graph of f have a horizontal tangent?
DERIVS. OF TRIG. FUNCTIONS
Example 2
Differentiate
For what values of x does the graph of f have a horizontal tangent?

18. The Quotient Rule gives:
DERIVS. OF TRIG. FUNCTIONS
Example 2
The Quotient Rule gives:

19. Since sec x is never 0, we see that f '(x) = 0 when tan x = 1.
DERIVS. OF TRIG. FUNCTIONS
Example 2
Since sec x is never 0, we see that
f '(x) = 0 when tan x = 1.
This occurs when x = nπ + π/4, where n is an integer.

20. Find the 27th derivative of cos x.
DERIVS. OF TRIG. FUNCTIONS
Example 4
Find the 27th derivative of cos x.
The first few derivatives of f (x) = cos x are as follows:

21. Therefore, f (24)(x) = cos x
DERIVS. OF TRIG. FUNCTIONS
Example 4
We see that the successive derivatives occur in a cycle of length 4 and, in particular, f (n)(x) = cos x whenever n is a multiple of 4.
Therefore, f (24)(x) = cos x
Differentiating three more times, we have:
f (27)(x) = sin x

22. Find In order to apply Equation 2, , we first
DERIVS. OF TRIG. FUNCTIONS
Example 5
Find
In order to apply Equation 2, , we first
rewrite the function by multiplying and dividing by 7:

23. If we let θ = 7x, then θ → 0 as x → 0. So, by Equation 2, we have:
DERIVS. OF TRIG. FUNCTIONS
Example 5
If we let θ = 7x, then θ → 0 as x → 0. So, by Equation 2, we have:

24. Calculate . Example 6 DERIVS. OF TRIG. FUNCTIONS
We divide the numerator and denominator by x: by the continuity of cosine and Eqn. 2