Answer
The exact value of the trigonometric function $\cos 2\alpha $ is $\frac{527}{625}$.
Work Step by Step
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 $\frac{527}{625}$.
