Express Mod Z in terms of Theta

**Bhaaring** · May 15th 2010, 05:12 PM

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 $x^2 + (y - 1)^2 = 1$ . 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 $z = x+iy$ , lies on the circle C defined by $x^2 + (y - 1)^2 = 1$ . 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** · May 16th 2010, 12:43 AM

Originally Posted by Bhaaring

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

Hi

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

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

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

Expand and simplify

**Bhaaring** · May 17th 2010, 02:48 AM

Originally Posted by running-gag

Hi

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

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

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

Expand and simplify

Thanks for the Help!

Expaning and simplifying i get an expression for |z|

$|z| = 2sin \theta$

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

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

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

Cheers,

**running-gag** · May 17th 2010, 08:42 AM

Originally Posted by Bhaaring

$|z| = 2sin \theta$

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

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

Did i get this right?

Yes
You can also write $z = \sin 2\theta + i (1 - \cos 2\theta)$

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 ...

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