## Precalculus (6th Edition) Blitzer

The exact value of the provided expression is $\frac{4}{5}$.
Let us assume $\theta ={{\tan }^{-1}}\left( \frac{3}{4} \right)$. Then, $\tan \theta =\frac{3}{4}$ As we know that $\tan \theta$ is positive, $\theta$ lies in the first quadrant. Now, by using the Pythagorean theorem: \begin{align} & {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\ & {{r}^{2}}={{4}^{2}}+{{3}^{2}} \\ & r=\sqrt{16+9} \\ & r=5 \end{align} Then, the value of the given expression is \begin{align} & \cos \left[ {{\tan }^{-1}}\frac{3}{4} \right]=\cos \theta \\ & =\frac{x}{r} \\ & =\frac{4}{5} \end{align}