2.9 Derivatives of the Trigonometric Functions

Summary

The basic differentiation formulas for each of the trigonometric functions are introduced. Only the derivative of the sine function is computed directly from the limit definition. The derivatives of all the other trig functions are derived by using the general differentiation rules.

By the end of your studying, you should know:

The limit of sin(x)/x as x goes to 0.
How to use the limit above to compute the limit of related quotients.
The derivatives of sine, cosine, tangent, secant, cosecant, and cotangent.
How to find the derivative of any function involving a trig function and application of the general rules of differentiation.

On-screen applet instructions: Use the slider to let x → 0. What is the limit of sin(x)/x as x → 0?

Examples

Consider the picture below:

θ is the central angle of the circle, s is the arc intercepted by θ, and d is the chord defined by θ. Find

A pilot flying at 3 miles above the ground at 600 miles per hour sights the airport with a spotting scope. How fast must she turn the scope when the angle between the path and plane is 40o to keep the scope pointed at the airport?

A block at the end of a spring is stretched past its rest position and released. Its position at time t is given by the formula d(t) = 4cos(t). Find the velocity of the spring at time t. When does the block move fastest?