Chapter 1.6, Problem 42E

To determine

To explain: The value of given trigonometric expression.

Answer to Problem 42E

Thevalue of given expression is $\frac{9\sqrt{22}}{44}$ .

Explanation of Solution

Given information:The given expressionis $\tan \left({\sin }^{-1}\left(\frac{9}{13}\right)\right)$ .

Calculation:

Let the angle $\theta$ be ${\sin }^{-1}\left(\frac{9}{13}\right)$ . It is known that $\tan \theta =\pm \frac{\sin \theta }{\sqrt{1-{\sin }^2\theta }}$ .

Substitute ${\sin }^{-1}\left(\frac{9}{13}\right)$ for $\theta$ in the above equation.

  $\begin{array}{c}\tan \theta =\frac{\sin \left({\sin }^{-1}\frac{9}{13}\right)}{\sqrt{1-{\left[\sin \left({\sin }^{-1}\frac{9}{13}\right)\right]}^2}}\\ =\frac{\frac{9}{11}}{\sqrt{1-{\left(\frac{9}{13}\right)}^2}}\\ =\frac{\frac{9}{13}}{\sqrt{\frac{88}{100}}}\end{array}$

Further simplify.

  $\begin{array}{c}\tan \theta =\frac{\frac{9}{13}}{\frac{2\sqrt{22}}{13}}\\ =\frac{9\sqrt{22}}{44}\end{array}$

Therefore, the value of given expression is $\frac{9\sqrt{22}}{44}$ .