1. ## Complex equation

First of all you'll have to excuse any miss-spelling an incorrect expressions, I am a swede and am not to good regarding the math grammar in english..

Now for the problem at hand:

Solve the equation and answer in polar form r(cosv + isinv)

$\displaystyle z^4 = -i$

Thanx!

2. Hello DenOnde

Welcome to Math Help Forum!

Your English is excellent. It's certainly much better than my Swedish!
Originally Posted by DenOnde
First of all you'll have to excuse any miss-spelling an incorrect expressions, I am a swede and am not to good regarding the math grammar in english..

Now for the problem at hand:

Solve the equation and answer in polar form r(cosv + isinv)

$\displaystyle z^4 = -i$

Thanx!
If we write
$\displaystyle z = r(\cos\theta + i\sin\theta)$
then, using De Moivre's Theorem:
$\displaystyle z^4 = r^4(\cos\theta + i\sin\theta)^4$
$\displaystyle =r^4(\cos4\theta+i\sin4\theta)$
We now express $\displaystyle - i$ in polar form. On the Argand diagram it is represented by the point $\displaystyle (0,-1)$, so has modulus $\displaystyle 1$ and argument $\displaystyle -\pi/2$. So:
$\displaystyle -i=1\Big(\cos(-\pi/2)+i\sin(-\pi/2)\Big)$
So:
$\displaystyle z^4=-i$

$\displaystyle \Rightarrow r^4(\cos4\theta+i\sin4\theta)=\cos(-\pi/2)+i\sin(-\pi/2)$

$\displaystyle \Rightarrow \left\{\begin{array}{l} r^4\cos4\theta=-\cos(-\pi/2)\\ r^4\sin4\theta = -\sin(-\pi/2) \end{array}\right .$

$\displaystyle \Rightarrow \left\{\begin{array}{l} r=1\\ 4\theta=-\pi/2+2n\pi, n \in \mathbb{Z} \end{array}\right .$

So the values of $\displaystyle \theta$ are given by:
$\displaystyle 4\theta = -\pi/2, -\pi/2\pm2\pi,-\pi/2\pm4\pi, ...$
and its principle values (those between $\displaystyle -\pi$ and $\displaystyle \pi$) are given by:
$\displaystyle \theta = -5\pi/8, -\pi/8, 3\pi/8, 7\pi/8$