Question

z1=1+sqrt3 i
part 1: Find the four angles that define the fourth roots of the number z1.

part 2: What are the fourth roots of z1= sqrt3 +1?

Answer 1

Well, we need to transform this a+bi form into:
\[r(\cos \theta + isin \theta)\]Now hopefully you have conversions for these somewhere where you'd be able to do them on your own, but these are the two conversions between rectangular and polar that we'll use:
\[r = \sqrt{a ^{2}+b ^{2}}\]which you may recognize as pythagorean theorem/distance formula. The second is:
\[\tan \theta = \frac{ b }{ a }\]
So with those conversions, you think you'd be able to start off and get r for me?

Answer 2

r=2

Answer 3

Awesome. So now
\[\tan \theta = \frac{ \sqrt{3} }{ 1 }\]Do you know what angle for tangent gets sqrt(3)?

Answer 4

60 degrees

Answer 5

Sounds right to me. So now we have this:
\[2(\cos60+isin60)\] Now this is the polar form, but we need roots. So this is how we find the roots.
\[\sqrt[n]{r}[\cos (\frac{ \theta }{ n }+\frac{ 2k \pi }{ n })+isin(\frac{ \theta }{ n }+\frac{ 2k \pi }{ n })]\]

The k starts from 0 and goes up to n-1 where n is the root we want. So if we want 4th roots, we want to START from k = 0 and n = 4. So this is our first 4th root:
\[\sqrt[4]{2}(\cos15+isin15)\] To get the next 3 roots, make k = 1, 2, 3 and get your new angles :3

Answer 6

:/ wow thats confusing

Answer 7

Since you're dealing with degrees instead of radians, probably best to write it like:
\[\frac{ 360k }{ n }\]

Answer 8

Okay, do you understand how we got thefirst answer, with 4th root 2 and 15 degrees?

Answer 9

not really

Answer 10

Ah. Okay, well how about the way we got 2(cos60+isin60)?

Answer 11

yeah I understand that it's just this whole roots thing thats super confusing

Answer 12

Right. I tried to put it in formula form, but that's prob a bit much xD Alright, so whatever root we want, we take r to that root and divide the angles by the number root. So since we want 4th root, I took the 4th root of 2 and divided the angles by 4.

Answer 13

okay so for the next root angle thing, would I just divide 15 by 4?

Answer 14

Nope. the 4th root 2 and the 15 degree one is our first root. The next roots are where the whole 360k/n part comes in. Since we have a 4th root, n = 4 and that 360 thing becomes 90k. So basically, the next 3 roots we want are found by adding 90 degrees. So if we want the 2nd root we add 90 to get 105. The next root we add 90 again to get 195. The final root we add 90 again to get 285. Those are your 4 roots :3 the 4th root(2) stays the same.

Answer 15

I get what you are saying, but I'm not sure what numbers to punch into my calculator to get the next root.

Answer 16

You don't punch numbers in, just leave it all in degrees or radical form.
4th(2)(cos15+isin15)
4th(2)(cos105+isin105)
4th(2)(cos195+isin195)
4th(2)(cos285+isin285)
Those would be your answers.

Answer 17

ohhhh! Okay! Thanks! So for the next part I have r=2 and the modulus is 30 degrees, so what next?

Answer 18

What does it want you to do with it?

Answer 19

find the fourth roots of z1=sqrt3+i

Answer 20

Okay, cool, r = 2 modulus is 30. So I think it's best to keep it in radians, so pi/6. Up toyou, unless it forces you to use degrees xD But yeah, same process, we take the 4th root of r and divide the angles by 4.

Answer 21

okay so the first root is 4root2(cos(7.5)+isin(7.5)) right?

Answer 22

correct. Now we take 360 and divide it by the root number, which gives us 90 degrees again. So now keep adding90 degrees until you have all 4 :P

Answer 23

so the next roots will be 97.5, 187.5, and 277.5, right?

Answer 24

That's all there is to it.

Answer 25

yay! Thanks again! You're my life saver!

Answer 26

As long as you understand it then its all good xD