Introduction to Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles of a triangle if we are given the lengths of the sides. Inverse trigonometric functions can be used to determine what angle would yield a specific sine, cosine, or tangent value. 

To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. 

The inverse of sine is arcsine (denoted $\arcsin$), the inverse of cosine is arccosine (denoted $\arccos$), and the inverse of tangent is arctangent (denoted $\arctan$).

Note that the domain of the inverse function is the range of the original function, and vice versa. An exponent of $-1$ is used to indicate an inverse function. For example, if $f(x) = \sin x$,then we would write $f^{-1}(x) = \sin^{-1} x$ . Be aware that $\sin^{-1} x$ does not mean $\displaystyle{\frac{1}{\sin x}}$. The reciprocal function is $\displaystyle{\frac{1}{\sin x}}$, which is not the same as the inverse function.

For a one-to-one function, if $f(a) = b$, then an inverse function would satisfy $f^{-1}(b) = a$. However, the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number $0$. 

Sine and cosine functions within restricted domains

(a) The sine function shown on a restricted domain of $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$; (b) The cosine function shown on a restricted domain of $\left[0, \pi\right]$.

The graph of the sine function is limited to a domain of $[-\frac{\pi}{2}, \frac{\pi}{2}]$, and the graph of the cosine function limited is to $[0, \pi]$. The graph of the tangent function is limited to $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Tangent function within a restricted domain

The tangent function shown on a restricted domain of $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

These choices for the restricted domains are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next, instead of being divided into pieces by an asymptote.

Definitions of Inverse Trigonometric Functions

We can define the inverse trigonometric functions as follows. Note the domain and range of each function.

The inverse sine function $y = \sin^{-1}x $ means $x = \sin y$. The inverse sine function can also be written $\arcsin x$. 

$\displaystyle{y = \sin^{-1}x \quad \text{has domain} \quad \left[-1, 1\right] \quad \text{and range} \quad \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]}$

The inverse cosine function $y = \cos^{-1}x $ means $x = \cos y$. The inverse cosine function can also be written $\arccos x$.

$\displaystyle{y = \cos^{-1}x \quad \text{has domain} \quad \left[-1, 1\right] \quad \text{and range} \quad \left[0, \pi\right]}$

The inverse tangent function $y = \tan^{-1}x$ means $x = \tan y$. The inverse tangent function can also be written $\arctan x$.

$\displaystyle{y = \tan^{-1}x \quad \text{has domain} \quad \left(-\infty, \infty\right) \quad \text{and range} \quad \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)}$

Graphs of Inverse Trigonometric Functions

The sine function and inverse sine (or arcsine) function

The arcsine function is a reflection of the sine function about the line $y = x$.

To find the domain and range of inverse trigonometric functions, we switch the domain and range of the original functions. 

The cosine function and inverse cosine (or arccosine) function

The arccosine function is a reflection of the cosine function about the line $y = x$.

Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line $y = x$.

The tangent function and inverse tangent (or arctangent) function

The arctangent function is a reflection of the tangent function about the line $y = x$.

Summary

In summary, we can state the following relations:

• For angles in the interval $\displaystyle{\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]}$, if $\sin y = x$, then $\sin^{−1} x=y$.
• For angles in the interval $\displaystyle{\left[0, \pi\right]}$, if $\cos y = x$, then $\cos^{-1} x = y$.
• For angles in the interval $\displaystyle{\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)}$, if $\tan y = x$, then $\tan^{-1}x = y$.