# Complex problem

• May 26th 2006, 08:08 PM
chancey
Complex problem
How do I find the answer to e^i(pi/18) *without* using a calculator.

Thanks
• May 27th 2006, 05:47 PM
ThePerfectHacker
Quote:

Originally Posted by chancey
How do I find the answer to e^i(pi/18) *without* using a calculator.

Thanks

Use Euler's Formula thus,
$\displaystyle e^{i(\pi/18)}=\cos (\pi/18)+i\sin(\pi/18)$
All you need to do is find what (co)sine of $\displaystyle \pi/18$ is, you can do this by using the third angle identity,
$\displaystyle \cos 3\theta=4\cos^3 \theta-3\cos \theta$
Which reduces to solving a cubic equation.
• May 27th 2006, 06:13 PM
chancey
Quote:

Originally Posted by ThePerfectHacker
Use Euler's Formula thus,
$\displaystyle e^{i(\pi/18)}=\cos (\pi/18)+i\sin(\pi/18)$
All you need to do is find what (co)sine of $\displaystyle \pi/18$ is, you can do this by using the third angle identity,
$\displaystyle \cos 3\theta=4\cos^3 \theta-3\cos \theta$
Which reduces to solving a cubic equation.

That doesnt help because sine and cosine are made up of the very expression I am trying to solve. This problem has been bugging me for ages
• May 27th 2006, 06:28 PM
ThePerfectHacker
Quote:

Originally Posted by chancey
That doesnt help because sine and cosine are made up of the very expression I am trying to solve. This problem has been bugging me for ages

Since, $\displaystyle \cos (\pi/6)=\frac{\sqrt{3}}{2}}$
You have,
$\displaystyle \frac{ \sqrt{3} }{2}=4x^3-3x$
Maybe time later I will solve this cubic, I predict it will look messy.
• May 27th 2006, 07:27 PM
chancey
Ahh, I see. So there is no formula(s) for solving any complex exponential? What if I had something really messy like, $\displaystyle e^{0.155i}$?

And yes, when that cubic equation is solved it gives the correct answer