Chapter 7, Problem 30RE

To determine

The simplified value of the expression $\cos \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)$ .

Answer to Problem 30RE

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

Explanation of Solution

Given information:

The expression is $\cos \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)$ .

Formula used:

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

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

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

Cosec function and cosine function related as,

  $\cos \theta =\sqrt{1-{\left(\frac{1}{\csc \theta }\right)}^2}$

Calculation:

Consider the expression $\cos \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)$ .

Recall that cosec function and cosine function related as,

  $\cos \theta =\sqrt{1-{\left(\frac{1}{\csc \theta }\right)}^2}$

Using it,

  $\cos \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)=\sqrt{1-{\left(\frac{1}{\csc \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)}\right)}^2}$

Observe that the value $\left|\frac{5}{3}\right|\ge 1$ .

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

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

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

Simplify further,

  $\begin{array}{c}\sqrt{1-{\left(\frac{1}{\csc \left({\csc }^{-1}\left(\frac{5}{3}\right)\right)}\right)}^2}=\sqrt{1-{\left(\frac{1}{\frac{5}{3}}\right)}^2}\\ =\sqrt{1-\left(\frac{9}{25}\right)}\\ =\frac{4}{5}\end{array}$

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