Chapter 4, Problem 17CT

To determine

To Find:The exact value of mentioned trigonometric expression

  $\left(\cot \left(\arcsin \frac{3}{8}\right)\right)$ .

Answer to Problem 17CT

The exact value of trigonometric expression

  $\left(\cot \left(\arcsin \frac{3}{8}\right)\right)$ is $2.472$ .

Explanation of Solution

Given information:

Use right angle triangle theorem concept.

Formula used:

Pythagoras theorem that is $p^2+b^2=h^2$ .

Where $p$ is perpendicular, $b$ is base and $h$ is height.

And inverse trigonometric formula ${\tan }^{-1}\left(x\right)={\cot }^{-1}\left(\frac{1}{x}\right)$ ;

  $\cot \left({\cot }^{-1}x\right)=x$

Where $x$ is variable.

Also by right angle triangle,$\begin{array}{l}\tan x=\frac{p}{b}\\ x={\tan }^{-1}\frac{p}{b}\end{array}$

Where $p$ is perpendicular and $b$ is base.

Calculation:

Consider the statement “mentioned trigonometric expression”

Recall that by the Pythagoras formula on right angle triangle,

  $p^2+b^2=h^2$ ; Where $p$ is perpendicular, $b$ is base and $h$ is height.

Since perpendicular is $3$ and height is $8$ .

Also let base of right angle triangle be $b$ .

So by Pythagoras formula

  $\begin{array}{l}\sqrt{p^2+b^2}=h\\ \Rightarrow \sqrt{3^2+b^2}=8\\ \Rightarrow 3^2+b^2=64\\ \Rightarrow b^2=64-9\\ \Rightarrow b=\sqrt{64-9}\\ \Rightarrow b=\sqrt{55}\end{array}$

Where $b$ is base and $h$ is height.

Recall the trigonometric formula by right angle triangle.

  $\begin{array}{l}\tan x=\frac{3}{\sqrt{55}}\\ x={\tan }^{-1}\frac{3}{\sqrt{55}}\end{array}$

Now, recall the trigonometric formula of tan−1(x)=cot−1(1x) ,

Therefore

  ${\tan }^{-1}\left(\frac{3}{\sqrt{55}}\right)={\cot }^{-1}\left(\frac{\sqrt{55}}{3}\right)$

  $={\sin }^{-1}\left(\frac{3}{8}\right)$

Recall the trigonometric formula of cot(cot−1x)=x

So,

  $\cot \left({\sin }^{-1}\left(\frac{3}{8}\right)\right)$ = $\cot \left({\cot }^{-1}\frac{\sqrt{55}}{3}\right)=\frac{\sqrt{55}}{3}$

So, the exactvalue of trigonometric function $\cot \left(\arcsin \frac{3}{8}\right)=\frac{\sqrt{55}}{3}=2.472$ .