2. without using a calculator, find the exact value of: a) sin(37π/12)

Explanation:

Step1: Rewrite the angle

We know that $\frac{37\pi}{12}=3\pi+\frac{\pi}{12}$. And $\sin(A + 2k\pi)=\sin A$ for integer $k$, and $\sin(A+\pi)=-\sin A$. So $\sin\frac{37\pi}{12}=\sin(3\pi+\frac{\pi}{12})=-\sin\frac{\pi}{12}$.

Step2: Express $\frac{\pi}{12}$ as a difference

We know that $\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}$. Then, by the formula $\sin(A - B)=\sin A\cos B-\cos A\sin B$, we have $\sin\frac{\pi}{12}=\sin(\frac{\pi}{3}-\frac{\pi}{4})=\sin\frac{\pi}{3}\cos\frac{\pi}{4}-\cos\frac{\pi}{3}\sin\frac{\pi}{4}$.

Step3: Substitute known values

We know that $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. So $\sin\frac{\pi}{12}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}-\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}$.

Step4: Find the original value

Since $\sin\frac{37\pi}{12}=-\sin\frac{\pi}{12}$, then $\sin\frac{37\pi}{12}=-\frac{\sqrt{6}-\sqrt{2}}{4}=\frac{\sqrt{2}-\sqrt{6}}{4}$.

Answer:

$\frac{\sqrt{2}-\sqrt{6}}{4}$