Limits on Trigonometric Functions

A fundamental trigonometric limit is the limit of sine theta divided by theta as theta approaches zero, which equals one.

To visualize, consider a unit circle and a small angle theta measured from the positive x-axis.

An arc corresponding to this angle is drawn. For a unit circle, the arc length is numerically equal to the angle in radians.

A vertical line is constructed from the arc's endpoint to the x-axis. This line segment represents the sine of theta. As the line BC is always smaller than arc AB, a relation sine theta over theta less than one is derived.

The radius is extended from the center O through point B until it intersects the tangent line at point A at a point D. The length of the line segment AD represents tan theta. Geometrically, the arc length theta is less than the length of the tangent segment AD, which can be rewritten in terms of sine and cosine theta.

Using both relations, the inequality is derived. As theta approaches zero, the outerbound values converge to one. Since the ratio of sine to theta lies between them, it also approaches one.