## Precalculus (6th Edition) Blitzer

The exact value of the provided expression is $-\frac{3}{4}$.
Let us assume $\theta ={{\cos }^{-1}}\left( -\frac{4}{5} \right)$. Then, $\cos \theta =-\frac{4}{5}$ As we know $\cos \theta$ is negative, $\theta$ lies in the second quadrant. Now, by using the Pythagorean theorem: \begin{align} & {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\ & {{5}^{2}}={{\left( -4 \right)}^{2}}+{{y}^{2}} \\ & y=\sqrt{25-16} \\ & y=3 \end{align} Then, the value of the given expression is \begin{align} & \tan \left[ {{\cos }^{-1}}\left( -\frac{4}{5} \right) \right]=\tan \theta \\ & =\frac{y}{x} \\ & =-\frac{3}{4} \end{align}