# Thread: Express Mod Z in terms of Theta

1. ## Express Mod Z in terms of Theta

Hey guys, i would appreciate a bit of help with my homework at the moment, this is a three part question taken from my math textbook (year 12). The first part (a.) was simple enough, sketch the circle defined by $\displaystyle x^2 + (y - 1)^2 = 1$. This was relatively simple, just a circle with an origin of (0,1) and a radius of 1.

These next few questions however have gotten me a little confused. Any help at all would be appreciated. I am not so much interested in the solution as i am with the method as i have several questions to do which are similar to this, so thorough explanation would be greatly appreciated if someone could manage this for me.

The point P(x,y) representing the non-zero complex number $\displaystyle z = x+iy$, lies on the circle C defined by $\displaystyle x^2 + (y - 1)^2 = 1$. Express Mod Z in terms of theta, the argument of Z.

Next, show that whatever the position of P on the circle C, the point P representing Z lies on a certain line, and determine the equation of this data line.

Thanks guys,

2. Originally Posted by Bhaaring
The point P(x,y) representing the non-zero complex number $\displaystyle z = x+iy$, lies on the circle C defined by $\displaystyle x^2 + (y - 1)^2 = 1$. Express Mod Z in terms of theta, the argument of Z.
Hi

$\displaystyle z = x + iy = |z| e^{i\theta} = |z| (\cos \theta + i \sin \theta)$

Therefore $\displaystyle x = |z| \cos \theta$ and $\displaystyle y = |z| \sin \theta$

Substitute into $\displaystyle x^2 + (y-1)^2 = 1$

Expand and simplify

3. Originally Posted by running-gag
Hi

$\displaystyle z = x + iy = |z| e^{i\theta} = |z| (\cos \theta + i \sin \theta)$

Therefore $\displaystyle x = |z| \cos \theta$ and $\displaystyle y = |z| \sin \theta$

Substitute into $\displaystyle x^2 + (y-1)^2 = 1$

Expand and simplify

Thanks for the Help!

Expaning and simplifying i get an expression for |z|

$\displaystyle |z| = 2sin \theta$

Substituting |z| into $\displaystyle z = |z| (\cos \theta + i \sin \theta)$

I get: $\displaystyle z = (2sin \theta) \cos \theta + (2sin \theta) i \sin \theta$

Did i get this right? or is there something else i need to do?

Cheers,

4. Originally Posted by Bhaaring
$\displaystyle |z| = 2sin \theta$

Substituting |z| into $\displaystyle z = |z| (\cos \theta + i \sin \theta)$

I get: $\displaystyle z = (2sin \theta) \cos \theta + (2sin \theta) i \sin \theta$

Did i get this right?
Yes
You can also write $\displaystyle z = \sin 2\theta + i (1 - \cos 2\theta)$

Originally Posted by Bhaaring
Next, show that whatever the position of P on the circle C, the point P representing Z lies on a certain line, and determine the equation of this data line.
This question is not clear to me ...