Chapter 6, Problem 9CLT

To determine

To find: Thevalue of the given trigonometric function.

Answer to Problem 9CLT

Thevalue of the trigonometric function is $\frac{3}{5}$ .

Explanation of Solution

Given:

The given trigonometric function is $\sin \left(\text{arctan}\frac{3}{4}\right)$ .

Calculation:

Consider the given expression is,

  $y=\sin \left(\text{arctan}\frac{3}{4}\right)$

As the value of the $\theta =\arctan \left(x\right)$ is $tan\theta =x$ then,

  $\begin{array}{l}tan\theta =\frac{3}{4}\\ \frac{\sin \theta }{\cos \theta }=\frac{3}{4}\\ 4\sin \theta =3\cos \theta \\ 4\sin \theta =3\sqrt{1-{\sin }^2\theta }\end{array}$

Solve further as,

  $\begin{array}{l}\frac{4\sin \theta }{3}=\frac{3\sqrt{1-{\sin }^2\theta }}{3}\\ \frac{4}{3}\sin \theta =\sqrt{1-{\sin }^2\theta }\\ \frac{16}{9}{\sin }^2\theta =1-{\sin }^2\theta \\ \frac{16}{9}{\sin }^2\theta +{\sin }^2\theta =1\end{array}$

Solve further as,

  $\begin{array}{l}{\sin }^2\theta =\frac{9}{25}\\ \sin \theta =\frac{3}{5}\end{array}$

Then, the value of the function is,

  $\begin{array}{c}y=\sin \left(\arctan \frac{3}{4}\right)\\ =\sin \left(\arctan \frac{3}{5}\right)\\ =\frac{3}{5}\end{array}$