# Express Mod Z in terms of Theta

#### Bhaaring

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,

#### running-gag

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

• Bhaaring

#### Bhaaring

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,

#### running-gag

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

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 $$\displaystyle z = \sin 2\theta + i (1 - \cos 2\theta)$$

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 ...