A student will be able to:
- Compute the derivatives of various trigonometric functions.
If the angle is measured in radians,
We can use these limits to find an expression for the derivative of the six trigonometric functions and . We first consider the problem of differentiating , using the definition of the derivative.
The derivative becomes
It will be left as an exercise to prove that
The derivatives of the remaining trigonometric functions are shown in the table below.
Derivatives of Trigonometric Functions
Keep in mind that for all the derivative formulas for the trigonometric functions, the argument is measured in radians.
It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.
Using the product rule and the formulas above, we obtain
Find if . What is the slope of the tangent line at ?
Using the quotient rule and the formulas above, we obtain
To calculate the slope of the tangent line, we simply substitute :
We finally get the slope to be approximately
If , find .
Substituting for ,
For examples of finding the derivatives of trigonometric functions (4.4), see Math Video Tutorials by James Sousa, The Derivative of Sine and Cosine (9:21).
Find the derivative of the following functions:
- If , find
- Use the definition of the derivative to prove that