3.14 Inverse trigonometric functions

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 :

$$f\left(x\right)={\sin }^{-1}x;x\in [-1,1]$$

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

$$\Rightarrow \sin \left\{f\left(x\right)\right\}=\sin \left\{{\sin }^{-1}\left(x\right)\right\}=x$$

In the representation of inverse function, we should treat “-1” as symbol – not as power. In particular,

$${\sin }^{-1}\left(x\right)\ne \frac{1}{\sin x}$$

Inverse trigonometric functions are also called arc functions. This is an alternative notation. The corresponding functions are arcsine, arccosine, arctangent etc. For example,

$$\Rightarrow f\left(x\right)={\sin }^{-1}\left(x\right)=\arcsin \left(x\right)$$

Nature of trigonometric functions

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! We shall see that we need to redefine trigonometric functions in order to make them invertible.

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.

We can identify many such shortened intervals for a particular trigonometric function. For example, the shortened domain of sine function can be any one of the intervals defined by :

$$[\left(2n-1\right)\frac{\pi }{2},\left(2n+1\right)\frac{\pi }{2}],n\in Z$$

The domain corresponding to n = 0 yields principal domain given by :

$$[-\frac{\pi }{2},\frac{\pi }{2}]$$

The nature of trigonometric functions is periodic. Same values repeat after certain interval. Here, our main task is to identify an interval of “x” such that all possible values of a trigonometric function are included once. This will ensure one-one relation in both directions between domain and range of the function. This interval is easily visible on graphs of the corresponding trigonometric function.

Inverse trigonometric functions

Every angle in the new domain (shortened) is related to a distinct real number in the range. Inversely, every real number in the range is related to a distinct angle in the domain of the trigonometric function. We are aware that the elements of the "ordered pair" in inverse relation exchanges their places. Therefore, it follows that domain and range of trigonometric function are exchanged for corresponding inverse function i.e. domain becomes range and range becomes domain.

Arcsine function

The arcsine function is inverse function of trigonometric sine function. From the plot of sine function, it is clear that an interval between $-\pi /2$ and $\pi /2$ 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.