# Limits and Derivatives of Trig Functions

If you graph and , you see that the graphs become almost indistinguishable near :

That is, as , . This approximation is often used in applications --- e.g. analyzing the motion of a simple pendulum for small displacements. I'll use it later on to derive the formulas for differentiating trig functions.

In terms of limits, this approximation says

(Notice that plugging in gives .) A derivation requires the Sandwich Theorem and a little geometry. What I'll give is not really a proof from first principles; you can think of it as an argument which makes the result plausible.

I've drawn a sector subtending an angle inside a circle of radius 1. (I'm using instead of x, since is more often used for the central angle.) The inner right triangle has altitude , while the outer right triangle has altitude . The length of an arc of radius 1 and angle is just .

(I've drawn the picture as if is nonnegative. A similar argument may be given if .)

Clearly,

Divide through by :

As , --- just plug in. By the Sandwich Theorem,

Taking reciprocals, I get

Example. Compute .

Plugging in gives . I have to do some more work.

The limit formula looks like this:

(I'm using instead of x to avoid confusing the variable in the formula with the variable in the problem.) The point is that the thing that is going to 0, the thing inside the sine, and the thing on the bottom must be identical.

In this problem, there is a inside the sine, but an x on the bottom. One or the other must change to match. I don't have nice ways of altering things inside a sine, but making the bottom into is easy:

Let . As , . So

Example. Compute .

Plugging in gives .

The idea here is to create terms of the form , to which I can apply my limit rule.

As , by the sine limit formula. , since and is continuous.

Example. Compute .

Plugging in gives . The limit may or may not exist.

Force the form to appear by using the trig identity :

Example. Compute .

If you draw the graph near with a graphing calculator or a computer, you are likely to get unusual results. Here's the picture produced by Mathematica:

The problem is that when x is close to 0, both and are very close to 0 --- producing overflow and underflow.

Actually, the limit is easy: Let . When , , so

For the last step, I used the result from the previous problem.

Example. Compute .

If you set , you get . Sigh.

I'll see what I can deduce by plotting the graph.

It looks as thought the limit is defined, and the picture suggests that it's around 3.5.

First, I'll break the tangents down into sines and cosines:

Next, I'll force the form to appear. Since I've got and , I need to make a and a to match:

Now take the limit of each piece:

The limit of a product is the product of the limits:

It's easy to derive the formulas for differentiating sine and cosine from the limit formula

and the angle addition formulas. I'll work out the formula for sine by way of example.

Let . Then

Now

Hence,

That is,

In similar fashion, you can derive the formula

Example.

Example. It's easy to derive the differentiation rules for the other trig functions from the ones for sine and cosine. Here are the formulas:

As an example, I'll derive the formula for cosecant. Remember that cosecant is the reciprocal of sine, so

Now you can use these formulas to compute derivatives involving these trig functions:

Example. For what values of x does have a horizontal tangent?

So where . In the range , this happens at . So for , where n is any integer.

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