Complex Trigonometric Solutions

[Bogrune]

Homework Statement

2.) Solve
x⁷=1
I am to solve for all the solutions to this polynomial, including the complex solutions using trigonometry. I'm supposed to plot my solutions on the unit circle, with the x-axis being the real axis, and the y-axis being the imaginary (or complex) axis.

Homework Equations

I'm supposed to use the following Identities to solve the equation: Rcis, meaning: R (real) times cosine + i(complex) sine.
cisⁿ(θ)=cis(nθ)

The Attempt at a Solution

I've numbered my steps taken in order:

x⁷=1
1.)(Rcisθ)⁷=1

2.)Rcis(θ)⁷=1

3.)cis⁷(θ)=1

4.)cis(7θ)=1

5.)cos(7θ) + isin(7θ)=1 + i0

5a.) cos(7θ)=1
7θ= cos^-1(1)
7θ= 0 + 2πn

5b.) sin(7θ)=0
7θ= sin^-1(0)


And that's where I got stuck.

 

[Stephen Tashi]

I've numbered my steps taken in order:

x⁷=1
1.)(Rcisθ)⁷=1

2.)Rcis(θ)⁷=1

3.)cis⁷(θ)=1

Do you have a "relevant equation" that explains why you can get rid of the R. (I don't use the cis notation, myself, so I'm not sure what you are supposed to say)

4.)cis(7θ)=1

5.)cos(7θ) + isin(7θ)=1 + i0

5a.) cos(5θ)=1

it should say cos(7θ) , of course

7θ= cos^-1(1)
7θ= 0 + 2πn

5b.) sin(7θ)=0
7θ= sin^-1(θ)

It should say sin^-1(0). (The solution won't add any new information because it will also imply that 7θ= 0 + 2πn

And that's where I got stuck.

Solve 7θ= 0 + 2πn for θ

Let n = 0,1,2... and you'll come back to where you started when n/7 = 1

 

[Bogrune]

I simply got rid of the "R" because it's not really a variable. I forgot to mention that "R" simply stands for "real", because of the fact that cosθ lies on the x-axis (or the real axis, in this case) and isinθ lies on the y (imaginary)-axis.
And sorry about the typos, I make those pretty often! ^^;

Solve 7θ= 0 + 2πn for θ

Let n = 0,1,2... and you'll come back to where you started when n/7 = 1

Oh, I think I see it now. So I plot those on the unit circle as θ= (0 +2πn)/7, and I get my final solution, right?

Though, thanks for the help!

 

[Stephen Tashi]

Yes, that's right. You are plotting what are called "the seventh roots of unity". In the complex numbers, 1 has 2 square roots, 3 cube roots, 4 fourth roots, etc.