Express Mod Z in terms of Theta

May 2010
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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,
 
Nov 2008
1,458
646
France
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
 
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May 2010
2
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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,
 
Nov 2008
1,458
646
France
\(\displaystyle |z| = 2sin \theta \)

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 ...