## Precalculus (6th Edition) Blitzer

The exact value of the trigonometric function $\tan 2\theta$ is $\frac{24}{7}$.
The figure shows the right-angle triangle. In this triangle, the base is $4$, the perpendicular is $3$, and the hypotenuse is $5$. Calculate the value of $\tan 2\theta$. Recall the double angle formula. \begin{align} & \tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta } \\ & =\frac{2\left( \frac{\text{perpendicular}}{\text{base}} \right)}{1-{{\left( \frac{\text{perpendicular}}{\text{base}} \right)}^{2}}} \end{align} Substitute $4$ for the base and $3$ for the perpendicular. \begin{align} & \tan 2\theta =\frac{2\left( \frac{\text{3}}{\text{4}} \right)}{1-{{\left( \frac{\text{3}}{\text{4}} \right)}^{2}}} \\ & =\frac{2\left( \frac{\text{3}}{\text{4}} \right)}{1-\frac{9}{16}} \\ & =\frac{\frac{\text{3}}{\text{2}}}{\frac{7}{16}} \\ & =\frac{24}{7} \end{align} Therefore, the exact value of the trigonometric function $\tan 2\theta$ is $\frac{24}{7}$.