Chapter 6, Problem 10CLT

To determine

The exact value of the expression without using acalculator.

Answer to Problem 10CLT

The exact value of the expression $\tan \left(\arcsin \frac{3}{5}\right)\text{ is }\frac{3}{4}$ .

Explanation of Solution

Given info:

  $\tan \left(\arcsin \frac{3}{5}\right)$

Formula used:

  

  $\begin{array}{l}r=\sqrt{x^2+y^2}\\ \text{sin}\theta \text{=}\frac{y}{r}\\ \text{cos}\theta \text{=}\frac{x}{r}\\ \text{tan}\theta \text{=}\frac{y}{x}\\ \text{csc}\theta \text{=}\frac{r}{y}\\ \text{sec}\theta \text{=}\frac{r}{x}\\ \text{cot}\theta \text{=}\frac{x}{y}\end{array}$

Results of inverse functions

  $\begin{array}{l}sin(\arcsin \theta )=\theta \\ \cos (\arccos \theta )=\theta \\ \tan (\arctan \theta )=\theta \end{array}$

Calculation:

We need to find $\tan \left(\arcsin \frac{3}{5}\right)$

We have

  $\begin{array}{l}y=3,r=5\\ 5=\sqrt{x^2+3^2}\\ x=4\\ \arcsin \frac{3}{5}=\arctan \frac{3}{4}\\ \tan \left(\arcsin \frac{3}{5}\right)=\tan \left(\arctan \frac{3}{4}\right)=\frac{3}{4}\\ \tan \left(\arctan \frac{3}{4}\right)=\frac{3}{4}\\ \tan \left(\arcsin \frac{3}{5}\right)=\frac{3}{4}\end{array}$

Conclusion:

Thus,the exact value of the expression $\tan \left(\arcsin \frac{3}{5}\right)\text{ is }\frac{3}{4}$ .