Finding the Polar Form of a Complex Number Using Euler's Relation

[aaj92]

Homework Statement

Using Euler's relation, prove that any complex number z=x+yi can be written in the form z= re^{i\theta} where r and \theta are real. Describe the significance of r and \theta with reference to the complex plane.

b) Write z= 3+4i in the form z = re^{i\theta}
(pretty sure I can get this one if I can get help on the proof.

Homework Equations

e^{i\theta}= cos\theta+isin\theta

The Attempt at a Solution

I tried to prove it, got what it wanted me to get but I feel like I did it wrong because I don't know how to go about doing part b. there's also a part c but I didn't feel the need to put it up here because if someone can just explain to me the proof for these equations I think I should be able to get parts b and c

 

[genericusrnme]

How do cartesian coordinates relate to polar coordinates?

 

[aaj92]

x = rcos\theta
y = rsin\theta

...is that all you have to do?

so that makes sense, but I guess I was wrong about knowing how to do part b then... I don't know how to find r and \theta given z = 3+4i

 

[genericusrnme]

yes
So you have your two equations
x=r\ Cos( \theta)
y=r\ Sin( \theta)

How would you find r in terms of x and y?

 

[aaj92]

r = \frac{x}{cos\theta}

r = i\frac{y}{sin\theta} ??

 

[genericusrnme]

r = \frac{x}{cos\theta}

r = i\frac{y}{sin\theta}


??

nono, r in terms of x and y does not contain any mention of \theta

Make use of the fact that Cos( \theta)^2 + Sin( \theta)^2 = 1.
You should end up with pythagoras' theorem.

To find \theta, you can make use of \frac{ Sin(\theta)}{Cos( \theta)} = Tan(\theta )

 

[aaj92]

ok well I'm lost :/

can't i just take the fact that x = rcos\theta and y= rsin\theta and plug that into z = x +iy? because that'll give the desired results right?

 

[aaj92]

oh... then i still don't know how to get part b. k well I'll have to figure the whole Pythagorean theorem thing out then

 

[genericusrnme]

ok well I'm lost :/

can't i just take the fact that x = rcos\theta and y= rsin\theta and plug that into z = x +iy? because that'll give the desired results right?

You can but that isn't going to help you find r and \theta
I'll show you how to find r, then I'll let you try and find \theta

1. I'm going to square both of our equations to get

x^2 = r^2 \ Cos(\theta )^2
y^2 = r^2 \ Sin(\theta )^2

2. Next I'm going to add these equations together

x^2 + y^2 = r^2 \ Cos(\theta )^2 + r^2 \ Sin(\theta )^2

3. I'm going to pull out a common factor of r^2

x^2 + y^2 = r^2 \ (Cos( \theta )^2 + Sin( \theta)^2 )

4. I now use the fact that Cos( \theta )^2 + Sin( \theta )^2 = 1 to find

x^2 + y^2 = r^2

5. Taking the square root of both sides

\sqrt{x^2 + y^2} = r

Which as I said before gives us pythagoras' theorem


So in b) you have z = 3 + 4i, we can now find the corresponding r, r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

All that's left now is to find \theta

 

[aaj92]

oh my god! thank you! I didn't know you could just add them together sorry my brain is just refusing to work right now but yeah I see how you can get theta now. thank you so much :)

 

[genericusrnme]

No problem buddy!