<s>[INST] 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.
 [/INST] One more method:
\begin{align*}
\left(\sin\frac{5\pi}{12}+\sin\frac{\pi}{12}\right)^2&=\left(\cos\frac{\pi}{12}+\sin\frac{\pi}{12}\right)^2\\
&=\cos^2\frac{\pi}{12}+\sin^2\frac{\pi}{12}+2\sin\frac{\pi}{12}\cos\frac{\pi}{12}\\
&=1+\sin\frac{\pi}{6}\\
&=\frac{3}{2}
\end{align*}
As $\displaystyle \sin\frac{5\pi}{12}>0$ and $\displaystyle \sin\frac{\pi}{12}>0$,
$$\sin\frac{5\pi}{12}+\sin\frac{\pi}{12}=\sqrt{\displaystyle \frac{3}{2}}=\frac{\sqrt{6}}{2}$$
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