Question

In Problems 79-90, find the exact value of each expression.

$\csc \left[2{\sin }^{-1}\left(-\frac{3}{5}\right)\right]$

Short answer

The exact value is$-\frac{25}{24}$

Step-by-step solution

Given information

Given expression$\csc \left[2{\sin }^{-1}\left(-\frac{3}{5}\right)\right]$

Rewriting using inverse rule and calculating 

Calculating, we get

$$\begin{gathered} \sin \left({\sin }^{-1}x\right)=x \\ {\sin }^{-1}x={\cos }^{-1}\left(\sqrt{1-x^2}\right) \\ a=\csc \left(2{\sin }^{-1}\left(-\frac{3}{5}\right)\right) \\ \theta ={\sin }^{-1}\left(-\frac{3}{5}\right) \\ \theta ={\cos }^{-1}\left(\frac{4}{5}\right) \end{gathered}$$

Using double angle formula, we geta=csc2sin-1-35a=12sinsin-1-35coscos-145a=12-3545a=-2524

Using, we get