There are several useful trigonometric limits that are necessary for evaluating the derivatives of trigonometric functions. Let's start by stating some (hopefully) obvious limits:

In order to evaluate the derivatives of
sine and
cosine we need to evaluate

Ifxis the measure of the central angle of a circle of radiusr, then the areaAof the sector determined byxis

A = r^{2}x/2

Let's start by looking at

If

(for the rest of this page, the arguments of the trig functions will be denoted by

If `A`_{1} is the area of the triangle
`AOP`, `A`_{2} is the area of the circular
sector `AOP`, and `A`_{3} is the area of the
triangle `AOQ`,

The area of a triangle is equal to one-half of the product of the base
times the height. Using this well-known result, and the above theorem
for the area of a sector of a circle (with `t` as the central
angle), we obtain:

From the Squeeze Theorem, it follows that

watko@mit.edu Last revised August 21, 1998.