# totally stuck

• Jul 31st 2008, 07:04 AM
DroMan
totally stuck
I have to find all complex number solutions and I have to leave my answer in trig form:

x^5-i=0

i have no idea how to start this problem. How would I go about getting the 1st degree to plug into the trig form?
• Jul 31st 2008, 07:26 AM
Soroban
Hello, DroMan!

Quote:

Solve: . $x^5-i\:=\:0$

We have: . $x^5 \:=\:i \quad\Rightarrow\quad x \:=\:i^{\frac{1}{5}}$

Convert to polar form: . $i \;=\;\cos\left(\frac{\pi}{2} + 2\pi n\right) + i\sin\left(\frac{\pi}{2} + 2\pi n\right)$

Use Demoivre's Theorem:

. . $i^{\frac{1}{5}} \;=\;\bigg[\cos\left(\frac{\pi}{2} + 2\pi n\right) + i\sin\left(\frac{\pi}{2} + 2\pi n\right)\bigg]^{\frac{1}{5}}$

. . . . $= \;\cos\left(\frac{\pi}{10} + \frac{2\pi}{5}n\right) + i\sin\left(\frac{\pi}{10} + \frac{2\pi}{5}n\right)\quad\hdots\quad \text{for }n \,=\,0,1,2,3,4$

Therefore: . $x \;=\;\begin{Bmatrix}\cos\frac{\pi}{10} + i\sin\frac{\pi}{10} \\ \\[-4mm] \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} &=& i \\ \\[-4mm] \cos\frac{9\pi}{10} + i\sin\frac{9\pi}{10} \\ \\[-4mm]
\cos\frac{13\pi}{10} + \sin\frac{13\pi}{10} \\ \\[-4mm] \cos\frac{17\pi}{10} + i\sin\frac{17\pi}{10}\end{Bmatrix}$

• Jul 31st 2008, 01:05 PM
DroMan
thnaks so much