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.

$$\sin\frac{5\pi}{12}+\sin\frac{\pi}{12}=\sqrt{\frac{1-\cos\frac{5\pi}{6}}{2}}+\sqrt{\frac{1-\cos\frac{\pi}{6}}{2}}=$$
$$=\sqrt{\frac{1+\frac{\sqrt3}{2}}{2}}+\sqrt{\frac{1-\frac{\sqrt3}{2}}{2}}=\frac{\sqrt{4+2\sqrt3}+\sqrt{4-2\sqrt3}}{2\sqrt2}=$$
$$=\frac{\sqrt3+1+\sqrt3-1}{2\sqrt2}=\sqrt{\frac{3}{2}}.$$
