Small-angle approximation

The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the basic trigonometric functions to a second-order approximation. This truncation gives:

where θ is the angle in radians.

The small angle approximation is useful in many areas of engineering and physics, including mechanics, electromagnetics, optics (where it forms the basis of the paraxial approximation), cartography, astronomy, and so on.

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes.

Figure 1. A comparison of the basic odd trigonometric functions to θ. It is seen that as the angle approaches 0 the approximations become better.

Figure 2. A comparison of cos θ to 1 − θ²/2. It is seen that as the angle approaches 0 the approximation becomes better.

The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the same length, meaning cos θ is close to 1 and θ²/2 helps trim the red away.

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