World Web Math: Derivatives of Sine

Suggested Prerequesites:

Definition of the Derivative, Useful Trigonometric Identities,

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:

Now we need to evaluate the limits and . We'll take an informal approach to evaluating these limits. The squeeze theorem would be used in a more formal approach. Let's look at the first of the two limits. Consider the graph of sin(h) :

Near h=0 it resembles the line y=x. It's slope 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.

The ratio of cos(h)-1 to h is about 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:

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 the tangent line to the graph at the point on the graph with x-coordinate

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page

jjnichol@mit.edu

Last modified 23 June 1997