Inverse Trigonometric Functions

A function f ( x ) has inverse if and only if it is one-to-one and onto. Thus, inverse of a function cannot be defined if it fails to be one-to-one. However, if we restrict the domain suitably, we can make the function to be one-to-one in the restricted domain. For example, y = x is not one-to-one for all real numbers.

But y = x is one-to-one and onto either for x ≥ or x ≤ . Hence y = x , x ≥ has the inverse f − ( x ) = √ x, x ≥ . Now, owing to their periodicity, none of six trigonometric functions is one-to-one over their natural domains. We shall restrict their domains so that trigonometric functions are one-to-one enabling the existence of their inverse functions.

This restriction can be done in many ways once again due to their periodicity. The conventional choices for the restricted domains are arbitrary but they have some important characteristics. Each restricted domain includes the number and some positive angles and the image of restricted domain contains the entire range. Let us define the inverse of sine function.

Consider f ( x ) = sin x, x ∈ [ − π , π ] . Then, sin x is one-to-one and onto in this restricted domain. Hence, the inverse of sine function exists. Note that f − ( y ) = x if and only if f ( x ) = y.

We write f − ( x ) = sin − ( x ) . Thus, inverse of sine is defined as sin − ( y ) = x if and only if sin x = y . Clearly, sin x : h − π , π i → [ − , ] and sin − x : [ − , ] → h − π , π i . Thus, sin − t is an angle whose sine is equal to t and which is located in h − π , π i .

Similarly we can define the other inverse trigonometric functions. Trigonometry The inverse functions sin − x, cos − x, tan − x, cosec − ( x ) , sec − ( x ) , cot − ( x ) are called inverse circular functions. For the function y = sin x , there are infinitely many angles x which satisfy sin x = t , − ≤ t ≤ . Of these infinite set of values, there is one which lies in the interval h − π , π i .

This angle is called the principal angle and denoted by sin − t . The principal value of an inverse function is that value of the general value which is numerically least. It may be positive or negative. When there are two values, one is positive and the other is negative such that they are numerically equal, then the principal value is the positive one.

We shall illustrate below the restricted domains, ranges of trigonometric functions and the domains, ranges of the corresponding inverse functions. (i) sin x : − π , π → [ − , ] ; sin − x : [ − , ] → h − π , π i (ii) cos x : [ , π ] → [ − , ] ; cos − x : [ − , ] → [ , π ] (iii) tan x : − π , π → ( −∞ , ∞ ) ; tan − x : ( −∞ , ∞ ) → − π , π (iv) cot x : ( , π ) → ( −∞ , ∞ ) ; cot − x : ( −∞ , ∞ ) → ( , π ) (v) cosec x : − π , π −{ } → R − ( − , ) ; cosec − x : R − ( − , ) → h − π , π i −{ } (vi) sec x : [ , π ] − π