Question

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

$\sin \left(2{\cos }^{-1}\frac{4}{5}\right)$

Short answer

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

Step-by-step solution

Given information

Given expression$\sin \left(2{\cos }^{-1}\frac{4}{5}\right)$

Assume θ=cos-1(45) and calculate sine

Calculating, we get

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

Substitute values in double angle formula

Using double angle formula, we get

$$\begin{gathered} \sin \left(2\theta \right)=2·\frac{3}{5}·\frac{4}{5} \\ \sin \left(2\theta \right)=\frac{24}{25} \end{gathered}$$