Lesson

For the sine function to have an inverse, we must restrict its domain to make it a one-to-one function. We restrict the domain of sine to the interval $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$[−π2,π2]. Its range is the usual range for sine, $[-1,1]$[−1,1].

It follows that the domain of the inverse sine function is the interval $[-1,1]$[−1,1] and its range is $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$[−π2,π2].

It was mentioned in a previous chapter that the graph of the inverse sine function, $\sin^{-1}x$`s``i``n`−1`x` has the same shape as the graph of the relation $x=\sin y$`x`=`s``i``n``y` when the domain of the sine function is suitably restricted. The graphs of $y=\sin x$`y`=`s``i``n``x` and $x=\sin y$`x`=`s``i``n``y` are displayed below.

The same discussion applies to the cosine function and its inverse. We restrict the domain of the cosine function to the interval $[0,\pi]$[0,π] over which the function is decreasing and hence, one-to-one. The range is the interval $[-1,1]$[−1,1].

The inverse cosine function, then, has domain $[-1,1]$[−1,1] and range $[0,\pi]$[0,π]. The graph of the function $\cos^{-1}x$`c``o``s`−1`x` is the same as the graph of $x=\cos y$`x`=`c``o``s``y`. The cosine function and its inverse are displayed below.

To obtain an inverse tangent function, we consider the tangent function restricted to the domain $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$(−π2,π2). The function is undefined at the endpoints, $-\frac{\pi}{2}$−π2 and $\frac{\pi}{2}$π2 where it has vertical asymptotes. The vertical asymptotes become horizontal asymptotes in the inverse function.

The inverse tangent function $\tan^{-1}x$`t``a``n`−1`x` is defined for all real numbers. Its range is the open interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$(−π2,π2). As before, we can obtain the graph of $\tan^{-1}x$`t``a``n`−1`x` by graphing $x=\tan y$`x`=`t``a``n``y`. The function $\tan x$`t``a``n``x` and its inverse are displayed below.

Draw the function $f(x)=\sin^{-1}x$`f`(`x`)=`s``i``n`−1`x`.

Draw the function $f(x)=3\cos^{-1}x$`f`(`x`)=3`c``o``s`−1`x`.

Draw the function $f(x)=\tan^{-1}2x$`f`(`x`)=`t``a``n`−12`x`.

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions