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Math Help - Express Mod Z in terms of Theta

  1. #1
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    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 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,
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  2. #2
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    Quote Originally Posted by Bhaaring View Post
    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
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  3. #3
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    Quote Originally Posted by running-gag View Post
    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,
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  4. #4
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    Quote Originally Posted by Bhaaring View Post
    |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)

    Quote Originally Posted by Bhaaring View Post
    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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