Inverse Trigonometric Functions

Inverse trigonometric functions reverse trigonometric operations and return the original angle. The inverse sine is often written as “arcsin” or “sine inverse”.

If a trigonometric function maps an angle to a ratio, its inverse function maps that ratio back to the original angle.

A function must be one-to-one to have an inverse. Otherwise, one output could match multiple inputs, causing ambiguity.

Since basic trigonometric functions are not inherently one-to-one, their domains are restricted to ensure this property.

For instance, sine is limited to negative pi over two to positive pi over two, setting the range for arcsin. Cosine is restricted from zero to pi, defining the range for arccos. These intervals ensure continuity, avoid repetition, and follow standard convention.

Tangent is limited to the same sine interval, but endpoints are excluded due to vertical asymptotes where it's undefined.

Graphically, inverse functions are reflections of their respective trigonometric functions across the line y = x.

These functions help find angles from given distances, such as calculating the elevation angle when the height and horizontal distance to an object are known.