Finding roots of equation.

[Crush.]

Hi there,

Need assistance answering the following question:

Find all roots of the equation Z^​8 = -2(√2) +2(√2)i

Express as r(cos(t)+isin(t) ?

and then

De Moivre's Theorem?


Many Thanks,

 

[pka]

Find all roots of the equation Z^​8 = -2(√2) +2(√2)i
Express as r(cos(t)+isin(t) ? and then
De Moivre's Theorem?

How about your showing what you have done on this?

 

[HallsofIvy]

Hi there,

Need assistance answering the following question:

Find all roots of the equation Z^​8 = -2(√2) +2(√2)i

Express as r(cos(t)+isin(t) ?

and then

De Moivre's Theorem?


Many Thanks,

Yes, that's exactly what you should do. Now, do it!

 

[Crush.]

Okay great, only thing is - I'm getting a little confused performing those steps.

I found r=|z| = √(-2(√2)² +2(√2)² = √8+8 = √16 = 4

So it'd end up..... 4[(cos(angle)+isin(angle)]

How do I find the angle? Help?

 

[pka]

Okay great, only thing is - I'm getting a little confused performing those steps.

I found r=|z| = √(-2(√2)² +2(√2)² = √8+8 = √16 = 4

So it'd end up..... 4[(cos(angle)+isin(angle)]

How do I find the angle? Help?

The angle is \(\displaystyle \dfrac{3\pi}{4}\). Do you see why?

 

[Crush.]

The angle is \(\displaystyle \dfrac{3\pi}{4}\). Do you see why?

Can you explain why that's the angle please?

 

[pka]

Can you explain why that's the angle please?

If the number is in II, the angle is \(\displaystyle \pi - \arctan\left(\left|\frac{y}{x}\right|\right)\)

 

[Crush.]

-

 

[HallsofIvy]

Okay great, only thing is - I'm getting a little confused performing those steps.

I found r=|z| = √(-2(√2)² +2(√2)² = √8+8 = √16 = 4

So it'd end up..... 4[(cos(angle)+isin(angle)]

How do I find the angle? Help?

4 cos(angle)+ i sin(angle)= 2√2+ 2√2i then cos(angle)= -√2/2 and sin(angle)= √2/2. In particular, that tells you that
\(\displaystyle tan(angle)= \frac{sin(angle)}{cos(angle)}= \frac{-\sqrt{2}/2}{\sqrt{2}{2}}= -1\)
Can you solve that for the angle?