Chapter 7, Problem 29RE

To determine

To Calculate: The simplified value of the expression $\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)$ .

Answer to Problem 29RE

The simplified value of the expression $\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)$ is $\frac{4}{5}$ .

Explanation of Solution

Given information:

The expression is $\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)$ .

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, $-1\le x\le 1$

Sine inverse function and cotangent inverse function related as,

  ${\cot }^{-1}x={\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$

Calculation:

Consider the expression $\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)$ .

Recall that sine inverse function and cotangent inverse function related as,

  ${\cot }^{-1}x={\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$

Using it,

  $\begin{array}{c}\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)=\sin \left({\sin }^{-1}\left(\frac{1}{\sqrt{1+{\left(\frac{3}{4}\right)}^2}}\right)\right)\\ =\sin \left({\sin }^{-1}\left(\frac{1}{\sqrt{\left(\frac{4^2+3^2}{4^2}\right)}}\right)\right)\\ =\sin \left({\sin }^{-1}\left(\frac{1}{\sqrt{\left(\frac{5^2}{4^2}\right)}}\right)\right)\\ =\sin \left({\sin }^{-1}\left(\frac{4}{5}\right)\right)\end{array}$

Observe that the value $\frac{4}{5}$ 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,

  $\sin \left({\sin }^{-1}\left(\frac{4}{5}\right)\right)=\frac{4}{5}$

Thus, the simplified value of the expression $\sin \left({\cot }^{-1}\left(\frac{3}{4}\right)\right)$ is $\frac{4}{5}$ .