Question

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

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

Short answer

The exact value is$\frac{24}{7}$

Step-by-step solution

Given information

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

Rewrite the inverse function and calculate the opposite side

Rewriting, we get

$$\begin{gathered} \theta ={\cos }^{-1}\left(-\frac{3}{5}\right) \\ c=\sqrt{5^2-3^2} \\ c=\sqrt{25-9} \\ c=\sqrt{16} \\ c=4 \end{gathered}$$

Calculate tangent and use double angle formula

Calculating, we get

$$\begin{gathered} \tan \theta =-\frac{4}{3} \\ \tan \left(2\theta \right)=\frac{2\left(-\frac{4}{3}\right)}{1-{\left(-\frac{4}{3}\right)}^2} \\ \tan \left(2\theta \right)=\frac{-\frac{8}{3}}{1-\frac{16}{9}} \\ \tan \left(2\theta \right)=\frac{-\frac{8}{3}}{-\frac{7}{9}} \\ \tan \left(2\theta \right)=\frac{24}{7} \end{gathered}$$