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.