## Precalculus (6th Edition) Blitzer

The exact value of the trigonometric function $\cos 2\alpha$ is $\frac{527}{625}$.
Calculate the value of the hypotenuse. For the right angle triangle. $\text{hypotenuse}=\sqrt{\text{perpendicula}{{\text{r}}^{2}}+\text{bas}{{\text{e}}^{2}}}$ Substitute $24$ for the base and $7$ for the perpendicular. \begin{align} & \text{hypotenuse}=\sqrt{{{\text{7}}^{2}}+\text{2}{{\text{4}}^{2}}} \\ & =\sqrt{625} \\ & =25 \end{align} Calculate the value of $\cos 2\alpha$. Recall the double angle formula. \begin{align} & \cos 2\alpha ={{\cos }^{2}}\alpha -{{\sin }^{2}}\alpha \\ & ={{\left( \frac{\text{base}}{\text{hypotenuse}} \right)}^{2}}-{{\left( \frac{\text{perpendicular}}{\text{hypotenuse}} \right)}^{2}} \end{align} Substitute $24$ for the base, $7$ for the perpendicular and $25$ for the hypotenuse. \begin{align} & \cos 2\alpha ={{\left( \frac{\text{24}}{\text{25}} \right)}^{2}}-{{\left( \frac{\text{7}}{\text{25}} \right)}^{2}} \\ & =\frac{576}{625}-\frac{49}{625} \\ & =\frac{527}{625} \end{align} Therefore, the exact value of the trigonometric function $\cos 2\alpha$ is $\frac{527}{625}$.