# Rewriting a trig function into a sum of exponential functions

Rewrite the function $2 + 4\sin(\pi t + \frac{\pi}{6})$ into a sum of exponential functions. By that I mean using Euler's formula $\sin(x) = \dfrac{e^{i\pi x} - e^{-i\pi x}}{2i}$.

If it wasn't for the $\frac{\pi}{6}$ term, this wouldn't be a problem for me, but I'm not sure what I can do to fix that.

• Everywhere you see '$x$', in the Euler formula you just place '$\pi t + \frac\pi 6$'. I think you are overcomplicating the question :) – jameselmore Sep 1 '15 at 18:42

Using additional formulas: $$2+4\sin(\pi t+\frac{\pi}{6})= 2+4\sin(\pi t)\cos\frac{\pi}{6}+4\cos(\pi t)\sin\frac{\pi}{6}=$$ $$=2+2\sqrt{3}\dfrac{e^{i\pi^2t} + e^{-i\pi^2t}}{2}+2\dfrac{e^{i\pi^2t} - e^{-i\pi^2t}}{2i}$$