New Zealand
Level 8 - NCEA Level 3

# Graphing Inverse Trigonometric Functions

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, $\sin^{-1}x$sin1x has the same shape as the graph of the relation $x=\sin y$x=siny when the domain of the sine function is suitably restricted. The graphs of $y=\sin x$y=sinx and $x=\sin y$x=siny 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 $\cos^{-1}x$cos1x is the same as the graph of $x=\cos y$x=cosy. 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 $\tan^{-1}x$tan1x 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 $\tan^{-1}x$tan1x by graphing $x=\tan y$x=tany. The function $\tan x$tanx and its inverse are displayed below.

#### Examples

##### QUESTION 1

Draw the function $f(x)=\sin^{-1}x$f(x)=sin1x.

##### QUESTION 2

Draw the function $f(x)=3\cos^{-1}x$f(x)=3cos1x.

##### QUESTION 3

Draw the function $f(x)=\tan^{-1}2x$f(x)=tan12x.

### Outcomes

#### M8-2

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