Chapter 6, Problem 1RE

To determine

To graph: The sketch of the inverse sine, cosine and tangent function and give the domain and range for each.

Answer to Problem 1RE

Solution: Domain and the range is:

$\begin{array}{|ccc|}\hline \text{function}& \text{domain}& \text{range}\\ {\sin }^{-1}x& \left[-1,1\right]& \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\\ {\cos }^{-1}x& \left[-1,1\right]& \left[0,\pi \right]\\ {\tan }^{-1}x& \left[-\infty ,\infty \right]& \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\\ \hline\end{array}$

Explanation of Solution

Graph: The graph of given function is:

First graph is inverse sine:

$2$ graphs are inverse cosine:

And, third graph is inverse tangent:

Interpretation: Domain and the range are:

$\begin{array}{|ccc|}\hline \text{function}& \text{domain}& \text{range}\\ {\sin }^{-1}x& \left[-1,1\right]& \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\\ {\cos }^{-1}x& \left[-1,1\right]& \left[0,\pi \right]\\ {\tan }^{-1}x& \left[-\infty ,\infty \right]& \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\\ \hline\end{array}$

In the question inverse sine, cosine and tangent lies between 1 and $-1$. So,

$\begin{array}{c}-1<{\cos }^{-1}x>1\\ -1{\sin }^{-1}x>1\\ -\infty {\tan }^{-1}x>\infty \end{array}$