Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 2 down vote accepted

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

share|cite|improve this answer
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 $e^{i\theta}=cos(\theta)+i\sin(\theta)$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.