## Precalculus (6th Edition) Blitzer

The exact value of the trigonometric function $\cos 2\theta$ is $\frac{7}{25}$.
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 $\cos 2\theta$. Recall the double angle formula. \begin{align} & \cos 2\theta ={{\cos }^{2}}\theta -{{\sin }^{2}}\theta \\ & ={{\left( \frac{\text{base}}{\text{hypotenuse}} \right)}^{2}}-{{\left( \frac{\text{perpendicular}}{\text{hypotenuse}} \right)}^{2}} \end{align} Substitute $4$ for the base, $3$ for the perpendicular and $5$ for the hypotenuse. \begin{align} & \cos 2\theta ={{\left( \frac{\text{4}}{\text{5}} \right)}^{2}}-{{\left( \frac{\text{3}}{\text{5}} \right)}^{2}} \\ & =\frac{16}{25}-\frac{9}{25} \\ & =\frac{7}{25} \end{align} Therefore, the exact value of the trigonometric function $\cos 2\theta$ is $\frac{7}{25}$.