Answer
The exact value of the trigonometric function $\sin \frac{\alpha }{2}$ is $\frac{\sqrt{2}}{10}$.
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 $\sin \frac{\alpha }{2}$.
Recall the half angle formula.
$\begin{align}
& \sin \frac{\alpha }{2}=\sqrt{\frac{1-\cos \alpha }{2}} \\
& =\sqrt{\frac{1-\left( \frac{\text{base}}{\text{hypotenuse}} \right)}{2}}
\end{align}$
Substitute $24$ for the base and $25$ for the hypotenuse.
$\begin{align}
& \sin \frac{\alpha }{2}=\sqrt{\frac{1-\left( \frac{\text{24}}{\text{25}} \right)}{2}} \\
& =\sqrt{\frac{1}{2\times 25}} \\
& =\frac{1}{5\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\
& =\frac{\sqrt{2}}{10}
\end{align}$
Therefore, the exact value of the trigonometric function $\sin \frac{\alpha }{2}$ is $\frac{\sqrt{2}}{10}$.
