# Using Argument and DeMoivre's theory

cutecarebear
Hello everyone,
I'm having a bit of trouble with using the argument function with DeMoivres formula. I have the question:
z8= 16
and am meant to find the solution using DeMoivre's formula (zn=rn(cosn(Θ) +isinn(Θ)) ). The problem is, I have no idea what an argument function is or how to find it. I've read around a bit and found that root(a2+b2)= r and that tan-1(b/a) =arg

but as you can see, I only have a (16) so that b=0, so is the argument then, tan-1(0)? I have several problems and they all have a but no b. However, in the answer section, they all have arguments. What the heck am I doing wrong?

## Homework Equations

I can do this one:
z3=-1 as it has no argument (though I don't know why), and is just a matter of plugging in numbers.
z3=-1
z3=1(cos(Θ)+isin(Θ)
z=3√(1)(cos(Θ)+isin(Θ))1/3
z=1(cos(2kπ)+isin(2kπ)1/3
z=1(cos(2kπ/3)+isin(2kπ/3)

and then, since it has no argument, k can equal 0, +/-1, +/-2, +/-3 etc..., you plug in the k and solve.

## The Attempt at a Solution

I haven't gotten very far with this one

z8=16
z8= 168(cos8(Θ)+isin8(Θ)) or if I do it the other way
z=8√(16)(cos(2kπ)+isin(2kπ))1/8
z=8√(16)(cos(2kπ/8)+isin(2kπ/8))

and I can't insert anything for k, because I don't know the argument. I know I sound like a real novice at math (I am), but I hope someone can help me! Thank you so much in advance!

## Answers and Replies

Imagine plotting a complex number z = x + i y in the complex plane, with the x coordinate the real part and the y coordinate the imaginary part. The argument is then the angle made between the real axis and a line drawn between the origin and the point z. Any positive real number has an argument of zero, and any negative real number has an argument of $$\pi$$. A positive imaginary number has an argument of $$\pi$$/2. So, don't say that 1 has "no argument", say that it has an argument of zero.

cutecarebear
Hey, thanks so much for your reply!

Using what you told me, I've managed to get to the next question, but I get stuck here. I'm not sure what I am doing wrong. Here's where I am so far:

z6=1+i
z=r1/6(cosΘ+isinΘ)1/6
z=6√2(cos(Θ/6)+isin(Θ/6))
z=6√2(cos(π/12)+isin(π/12))

and so

zk=6√2 ei(π+2πk/12)

The answer I need to get is:

zk=21/12ei(π+8πk/24)

I don't understand at all what happened. Am I still doing the argument incorrectly? Thanks in advance!

I think you're still doing the argument incorrectly. It looks like you have the right answer for the magnitude, since the sixth root of sqrt(2) = 21/12. But what is the argument of 1+i? You should get tan-1(1/1) = tan-1(1) = $$\pi$$/4