Without the aid of a calculator, compute $\sin\frac{5\pi}{12}+\sin\frac{\pi}{12}$ 
Without the aid of a calculator, compute $\sin\frac{5\pi}{12}+\sin\frac{\pi}{12}$

The only method in doing this that I know was to just plug it in into the calculator, but I really had no clue on how to do this without the aid of a calculator, 
I was thinking more along the lines of using the Taylor polynomial series but that really didn't work out for me though.

Use the product formula:
$$\sin(a)\cos(b) = \frac{1}{2}(\sin(a+b) + \sin(a-b))$$
let $a = \frac{3\pi}{12}$ and $b = \frac{2\pi}{12}$
Then 
$$\sin(\frac{5\pi}{12}) + \sin(\frac{\pi}{12})=
\sin(\frac{3\pi}{12}+\frac{2\pi}{12}) + sin(\frac{3\pi}{12}-\frac{2\pi}{12}) $$
$$=2\sin(\frac{3\pi}{12})\cos(\frac{2\pi}{12})=2\sin(\frac{\pi}{4})\cos(\frac{\pi}{6})$$
