World Web Math: Derivatives of Trigonometric Functions

Definition of the Derivative, Introduction to Trigonometric Functions, Useful Trigonometric Identities, Evaluating Limits

The trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are used in many applications of calculus. We'd like to be able to find their derivatives. Let's start with sine, everyone's favorite trig function, and apply the definition of the derivative:

Instead of a formal evaluation of the limit, instead consider the graph of sin(h). It's slope near h=0 is very close to 1. The ratio of sin(h) to h near h=0 is also very close to one. Therefore, it's not outrageous (and is in fact correct) to guess that

We can apply the same reasoning to The graph of cos(h)-1 is flat at h=0, and has a slope of zero. Therefore, we guess (again correctly) that

Now we can finish evaluating :

Now, step back a minute a think about this result for a minute. The slope of sine is zero whenever cosine is zero. Sine is increasing where cosine is positive, and decreasing where cosine is negative. So, this makes sense.

Exercises:

Find the derivatives of the following functions:
3. If , find

   a) the x-coordinates of all points on the graph where the tangent line is parallel to the line

   b) an equation of teh tangent line to the graph at teh point on the graph with x-coordinate