# Inverse trigonometric and hyperbolic functions

**Understanding Inverse Trigonometric and Hyperbolic Functions**

**Inverse trigonometric functions**are the reciprocal relationships of the basic trigonometric functions: sine, cosine and tangent. They are denoted as arcsin, arccos, and arctan.- There are also cosecant, secant, and cotangent inverse trigonometric functions, although these are less commonly used.
- Inverse trigonometric functions allow you to solve for the angle when given a ratio of sides in a right triangle. For example, if given an opposite side and an adjacent side, we can find the angle using the inverse tangent function.
**Hyperbolic functions**are analogues of the ordinary trigonometric or circular functions.- The basic hyperbolic functions are hyperbolic sine ‘sinh’, and hyperbolic cosine ‘cosh’.
- There are also hyperbolic tangent (tanh), hyperbolic cosecant (csch), hyperbolic secant (sech), and hyperbolic cotangent (coth).

**Properties of Inverse Trigonometric Functions**

- Inverse trigonometric functions will return the radian measure of the angle for a given ratio.
- The arcsin function allows you to solve for the angle when given the ratio of opposite over hypotenuse.
- The arccos function finds the angle when given the ratio of adjacent over hypotenuse.
- The arctan function provides the angle for a given ratio of opposite over adjacent.

**Properties of Hyperbolic Functions**

- Hyperbolic functions behave in some ways similar to their trigonometric counterparts, but have different identities.
- For example, while it is generally true for trigonometric functions that cos^2(x) + sin^2(x) = 1, for hyperbolic functions, cosh^2(x) - sinh^2(x) = 1.
- The derivatives and integrals of hyperbolic functions are noteworthy and follow similar patterns to their trigonometric counterparts.

**Applications of Inverse Trigonometric and Hyperbolic Functions**

- Inverse trigonometric functions are used in a variety of mathematical and real-world applications including physics, engineering, and computer science to find the angle from a specific tangent, or solve for unknown angles in right triangles.
- Hyperbolic functions have many uses in physics e.g. modeling behaviour of hanging cables (catenary curves) or modeling population growth, as well as in complex numbers and powers of matrices.