## Precalculus (6th Edition) Blitzer

The exact value of the provided expression is $\frac{4}{5}$.
Let us assume $\theta ={{\cos }^{-1}}\left( \frac{3}{5} \right)$. Then, $\cos \theta =\frac{3}{5}$ As $\cos \theta$ is positive, $\theta$ lies in the first quadrant. Now, by using the Pythagorean theorem: \begin{align} & {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\ & {{5}^{2}}={{3}^{2}}+{{y}^{2}} \\ & y=\sqrt{25-9} \\ & y=4 \end{align} Then, the value of the given expression is \begin{align} & \sin \left[ {{\cos }^{-1}}\frac{3}{5} \right]=\sin \theta \\ & =\frac{y}{r} \\ & =\frac{4}{5} \end{align}