Inverse Trigonometric Functions

Introduction

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. The sine function takes an angle and tells the length of the y-component (rise) of that triangle. The cosine function takes an angle and tells the length of x-component (run) of a triangle. The tangent function takes an angle and tells the slope (y-component divided by the x-component).

Definition

Inverse trigonometric function returns an angle corresponding to a real number, following certain rule. They are inverse functions corresponding to trigonometric functions. The inverse function of sine , for example, is defined as :

f(x) =sin^-1x; x ∈ [-1,1]

where x is a real number, f(x)is the angle. Clearly, f(x) is the angle, whose sine is x.

Trigonometric functions are many-one relations. The trigonometric ratio of different angles evaluate to same value. If we draw a line parallel to x-axis such that 0 < y < 1, then it intersects sine plot for multiple times ,in fact, infinite times. It follows, then, that we can associate many angles to the same sine value. The trigonometric functions are, therefore, not an injection and hence not a bijection. As such, we can not define an inverse of trigonometric function in the first place

In order to define, an inverse function, we require to have one-one relation in both directions between domain and range. The function needs to be a bijection. It emerges that we need to shorten the domain of trigonometric functions such that a distinct angle corresponds to a distinct real number. Similarly, a distinct real number corresponds to a distinct angle.

Inverse of sin function

The arcsine function is inverse function of trigonometric sine function. An interval between -π/2 and π/2 includes all possible values of sine function only once. Note that end points are included. The redefinition of domain of trigonometric function, however, does not change the range.

Inverse of cos function

The arccosine function is inverse function of trigonometric cosine function. An interval between 0 and π includes all possible values of cosine function only once. Note that end points are included. The redefinition of domain of trigonometric function, however, does not change the range.

Inverse of tan function

The arctangent function is inverse function of trigonometric tangent function. An interval between -π/2 and π/2 includes all possible values of tangent function only once. Note that end points are excluded. The redefinition of domain of trigonometric function, however, does not change the range.

These are the main inverse trigonometric functions. The other functions are also defined according to the trigonometric base functions.