Convert sqrt(27) + 3i to Re^{iθ}: Guide

[Lancelot59]

The problem is that I need to convert:

[tex] \sqrt{27} + 3i[/tex]

From the form [tex](a+bi)[/tex] to [tex]Re^{i\theta}[/tex]. I have no clue what to do with this. I do know the following:

[tex]e^{i\pi}=cos(\theta)+sin(\theta)i=-1[/tex]

But I don't see how that's helpful. This is the first of several problems.

 

[SVXX]

What you need to know is that for a complex number z = a + ib, we define the following -:

and

Now can you complete this question?

Sidenote : r is called the "modulus" or "magnitude" of the complex number and theta is its "amplitude" or the "argument".

 

[Lancelot59]

What you need to know is that for a complex number z = a + ib, we define the following -:

and

Now can you complete this question?

Sidenote : r is called the "modulus" or "magnitude" of the complex number and theta is its "amplitude" or the "argument".

I see here. Correct me if I'm wrong here:

An imaginary number is like a vector, with a real and imaginary component. Am I correct in saying that the new vector:
[tex]6e^{i\frac{\pi}{6}}[/tex]
Is now in "polar coordinates"? Or some form of polar coordinates with an imaginary axis?

Another question, is modulus just a term used for the magnitude, or is there more to it than that? I've always known it to be the remainder of division.

 

[SVXX]

Indeed, it is in polar coordinates in the exponential form.

And

which is why it is called the "modulus" of a complex number (its how the modulus function is defined for complex numbers).

 

[Lancelot59]

Indeed, it is in polar coordinates in the exponential form.

And

which is why it is called the "modulus" of a complex number (its how the modulus function is defined for complex numbers).

Okay, now I have one where I need to go the other direction:

[tex]2e^{\frac{2\pi}{3}i}[/tex]

So I got the following:

[tex]tan(\frac{2\pi}{3})=\frac{b}{a}=-\sqrt{3}[/tex]
[tex]b=-a\sqrt{3}[/tex]

[tex]\sqrt{a^{2}+b^{2}}=2[/tex]
[tex]a^{2}+b^{2}=4[/tex]
[tex]a^{2}+(-a\sqrt{3})^{2}=4[/tex]
[tex]a^{2}+3a^{2}=4[/tex]
[tex]4a^{2}=4[/tex]
[tex]a^{2}=1[/tex]
[tex]a=+-1[/tex]

I'm not sure where to go from here.

EDIT: Nevermind, I was looking at the wrong answer in the answer key.