Complex Fourth Root

[greatwhiteshark]

I have two questions below. I will show partial work for each question.
I will NOT type the complex root formula (too BIG). I will display my answers per question. Can someone please check to see if I found the right complex roots for each question?

1) Find the complex fourth root of -2 + 3i.

n = 4

k = 0, 1, 2, and 3 = four roots

radius = sqrt{13}

I am in quadrant 2 where my reference angle = 3/-2, right?

Theta = 180 degrees - my reference angle, right?

Thus, theta = 179.974 degrees. Is this correct so far?

My Answers: d = degrees

First root = 13^1/4 (cos44.994d + i sin 44.994d)

Second root: 13^1/4 (cos134.994d + i sin 134.994d)

Third root: 13^1/4 (cos 224.994d + i sin224.994d)

Fourth root: 13^1/4 (cos314.994d + i sin314.994d)

Is this correct? If not where did I get lost?

Here's my second question:

2) Find the complex cube root of -1 +sqrt{3}i

n = 3

k = 0, 1 and 2

radius = 2

I am in the second quadrant, right? So, my reference angle = -60 degrees. Here theta = 120 degrees = the initial angle, right?

MY ANSWERS: d = degrees

First root: 2^1/3 (cos40d + i sin40d)

Second root: 2^1/3 (cos160d + i sin160d)

Third root: 2^1/3 (cos 280d + i sin280d)

Is this correct? If not, where did I go wrong?

 

[Guest]

1.

I am in quadrant 2 where my reference angle = 3/-2, right?

The <u>tangent</u> of the reference angle is 3/2. Negatives are ignored when you're finding the reference angle. So the reference angle is tan^-1(3/2).

Theta = 180 degrees - my reference angle, right?

Right.

Thus, theta = 179.974 degrees.

No. I'm not sure where you got that.

Once you find theta, you've got the right idea for finding the roots.

2.

my reference angle = -60 degrees.

Usually you'd just say it's 60 degrees.

The rest of number 2 is right.

A comment on notation:

13^1/4 = (13^1) / 4

You mean 13^(1/4). Likewise for 2^(1/3).

 

[tkhunny]

I am in quadrant 2 where my reference angle = 3/-2, right?

No. The tangent of your reference angle is -3/2, but don't worry about the sign.

Thus, theta = 179.974 degrees.

Where did you get that? That is very close to 180­°! Where is this point? It is not very close to the negative x-axis. Try again with the tangent hint given above.

Edit: Alex beats me to it...

 

[greatwhiteshark]

alexflury

alexflury

Thank you for your notes.