Answer
The exact value of the trigonometric function $\cos 2\theta $ is $\frac{7}{25}$.
Work Step by Step
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 $\frac{7}{25}$.
