Differentiation of Trigonometric Functions - Proofs of Derivative of The Sine and Cosine Functions - Limit of As

Limit of As

The diagram on the right shows a circle, centre O and radius r. Let θ be the angle at O made by the two radii OA and OB. Since we are considering the limit as θ tends to zero, we may assume that θ is a very small positive number: 0 < θ ≪ 1.

Consider the following three regions of the diagram: R₁ is the triangle OAB, R₂ is the circular sector OAB, and R₃ is the triangle OAC. Clearly:

Using basic trigonometric formulae, the area of the triangle OAB is

The area of the circular sector OAB is, while the area of the triangle OAC is given by

Collecting together these three areas gives:

Since r > 0 we can divide through by ½·r2; this means that the construction and calculations are all independent of the circle's radius. Moreover, since 0 < θ ≪ 1 it follows that sin(θ) > 0 and we may divide through by a factor of sin(θ), giving:

In the last step we simply took the reciprocal of each of the three terms. Since all three terms are positive this has the effect of reversing the inequities, e.g. if 2 < 3 then ½ > ⅓.

We have seen that if 0 < θ ≪ 1 then sin(θ)/θ is always less than 1 and, in addition, is always greater than cos(θ). Notice that as θ gets closer to 0, so cos(θ) gets closer to 1. Informally: as θ gets smaller, sin(θ)/θ is "squeezed" between 1 and cos(θ), which itself it heading towards 1. It follows that sin(θ)/θ tends to 1 as θ tends to 0 from the positive side.

For the case where θ is a very small negative number: –1 ≪ θ < 0, we use the fact that sine is an odd function: