Chapter 7, Problem 39RE

To determine

The simplified value of the expression $\sin \left({\csc }^{-1}u\right)$ in terms of $u$ .

Answer to Problem 39RE

The simplified value of the expression $\sin \left({\csc }^{-1}u\right)$ in terms of is $\frac{1}{u}$ .

Explanation of Solution

Given information:

The expression is $\sin \left({\csc }^{-1}u\right)$ .

Formula used:

For the sine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Inverse of sine and inverse of cosec related as,

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

Where, $\left|x\right|\ge 1$ .

Calculation:

Consider the expression $\sin \left({\csc }^{-1}u\right)$ .

Recall that inverse of sine and inverse of cosec related as,

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

Where, $\left|x\right|\ge 1$ .

Applied it,

  $\sin \left({\csc }^{-1}u\right)=\sin \left({\sin }^{-1}\frac{1}{u}\right)$

Where, $\left|u\right|\ge 1$ .

Observe that the value $\frac{1}{u}$ is lies between $-1$ and $1$ .

Recall that for the sine function and its inverse, the following holds:

  $\begin{array}{c}f\left(f^{-1}\left(x\right)\right)=\sin \left({\sin }^{-1}\left(x\right)\right)\\ =x\end{array}$

Where, $-1\le x\le 1$

Simplify further,

  $\sin \left({\sin }^{-1}\frac{1}{u}\right)=\frac{1}{u}$

Thus, the simplified value of the expression $\sin \left({\csc }^{-1}u\right)$ is $\frac{1}{u}$ .