Question

if \\( \sin \frac{\pi}{12}=\frac{1}{2} \sqrt{a-\sqrt{b}} \\), then, by using a half - angle formula, find \\( a= \\) \\( b= \\)

Explanation:

Step1: Apply the half - angle formula

The half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$.
Let $\frac{\alpha}{2}=\frac{\pi}{12}$, then $\alpha=\frac{\pi}{6}$.
So, $\sin\frac{\pi}{12}=\sqrt{\frac{1-\cos\frac{\pi}{6}}{2}}$ (since $\frac{\pi}{12}$ is in the first quadrant and sine is positive there).

Step2: Substitute the value of $\cos\frac{\pi}{6}$

We know that $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$.
Substitute it into the formula:
$\sin\frac{\pi}{12}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{3}}{4}}=\frac{1}{2}\sqrt{2-\sqrt{3}}$

Answer:

$A = 2$, $B = 3$