# Complex Exponential Expansion

I'm currently working through a problem using exponentials with complex numbers! I am simply wondering what the steps are to go from

$$\pi \cdot (e^{i\frac{\pi}{3}} - e^{i\pi}) = \frac{\pi}{2}\cdot(3 + i\sqrt{3})$$

If someone could point me in the direction of the correct expansion that would be very helpful!

Thank you

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Dear Sarah, first of all, you may cancel the factor of $\pi$ on both sides.

Second, $\exp(ix)=\cos(x)+i\sin(x)$ and $$\cos(\pi/3)=\frac 12, \quad \sin(\pi/3) = \frac{\sqrt{3}}{2}, \quad e^{i\pi} = -1$$ so $$e^{i\pi/3} - e^{i\pi} = \frac 12 + i \frac{\sqrt{3}}{2} + 1 = \frac 32 + i\frac{\sqrt{3}}{2} = \frac{1}{2} (3 +\sqrt{3})$$ just like you wrote. If you need it, $\cos(\pi/3)$ is $\cos (60°)$ - thanks, Hans - which is equal to $1/2$ because of a simple picture of an equilateral triangle. $\sin(\pi/3)$ is calculated via $\sin^2x+\cos^2 x = 1$. For a derivation of the exponential of an imaginary number in terms of cosines and sines, see e.g.

Complex Exponents

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Thats brilliant, thanks for your help! –  Sarah JT May 19 '11 at 12:08
It's 60 degrees, not 120. –  Hans Lundmark May 19 '11 at 17:00
There is an $i$ missing after the last equality sign. –  Raeder May 21 '11 at 6:05
Hint: Use $e^{i\theta}=cos(\theta)+i\sin(\theta)$.