use a half - angle formula to evaluate the expression without using a calculator. cos(-5π/12) f. choose the…

Explanation:

Step1: Recall the half - angle formula for cosine

The half - angle formula for $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. We want to find $\cos(-\frac{5\pi}{12})=\cos(\frac{5\pi}{12})$. Let $\frac{\alpha}{2}=\frac{5\pi}{12}$, then $\alpha=\frac{5\pi}{6}$.

Step2: Determine the sign

Since $\frac{\pi}{2}<\frac{5\pi}{12}<\pi$, $\cos(\frac{5\pi}{12})<0$.

Step3: Substitute $\alpha$ into the formula

We know that $\cos\alpha=\cos(\frac{5\pi}{6})=-\frac{\sqrt{3}}{2}$. Substituting into the half - angle formula $\cos(\frac{5\pi}{12})=-\sqrt{\frac{1+\cos(\frac{5\pi}{6})}{2}}=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=-\sqrt{\frac{2 - \sqrt{3}}{4}}=-\frac{\sqrt{2-\sqrt{3}}}{2}$.

For the multiple - choice part: We know that $\cos(\frac{5\pi}{12})=-\sqrt{\frac{1+\cos(\frac{5\pi}{6})}{2}}$. Since $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$, then $\cos(-\frac{5\pi}{12})=\cos(\frac{5\pi}{12})=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}$. The correct formula is B. $\cos(-\frac{5\pi}{12})=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}$, and the numerator of the expression under the radical is $1-\frac{\sqrt{3}}{2}$.

Answer:

B. $\cos(-\frac{5\pi}{12})=-\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}$