use the adjacent figure to find the exact value of the following trigonometric function. cos(α/2) cos(α/2)=□…

Explanation:

Step1: Find the hypotenuse

By the Pythagorean theorem, for a right - triangle with legs $a = 3$ and $b = 4$, the hypotenuse $c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5$. So, $\cos\alpha=\frac{3}{5}$.

Step2: Use the half - angle formula for cosine

The half - angle formula for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}$. Since $\alpha$ is an acute angle (as it is in a right - triangle), $\frac{\alpha}{2}$ is also acute and $\cos\frac{\alpha}{2}>0$. Substitute $\cos\alpha=\frac{3}{5}$ into the formula: [ \begin{align*} \cos\frac{\alpha}{2}&=\sqrt{\frac{1+\frac{3}{5}}{2}}\ &=\sqrt{\frac{\frac{5 + 3}{5}}{2}}\ &=\sqrt{\frac{\frac{8}{5}}{2}}\ &=\sqrt{\frac{8}{5}\times\frac{1}{2}}\ &=\sqrt{\frac{4}{5}}\ &=\frac{2}{\sqrt{5}}\ &=\frac{2\sqrt{5}}{5} \end{align*} ]

Answer:

$\frac{2\sqrt{5}}{5}$