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Homework Help: Math problem

  1. Nov 14, 2003 #1
    hope to get the idea on how to solve this question.

    the complex number z is given by

    z = 1 + cos (theta) + i sin (theta)

    where -pi < theta < or = +pi


    show that for all values of theta, the point representing z in a Argand diagram is located on a circle. find the centre and radius of the circle.
     
  2. jcsd
  3. Nov 14, 2003 #2

    HallsofIvy

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    If z = 1 + cos (&theta;) + i sin (&theta)

    Then z-1= cos(&theta)+ i sin(&theta;).

    If you represent z as x+ iy then
    (x-1)+ iy= cos(&theta;)+ i sin(&theta;)

    or x- 1= cos(&theta;), y= sin(&theta;)

    Those are parametric equations of a circle with what center and radius?
     
  4. Nov 15, 2003 #3
    ok. i compare those with the
    y=r sin (&theta)
    and
    x=r cos (&theta)


    so, i know the radius = 1 unit
    but may i know how to find the centre of the circle?

    tq.
     
  5. Nov 15, 2003 #4

    HallsofIvy

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    It's exactly where the center of the circle given by

    x= r cos &theta;
    y= r sin &theta; is!

    Hint: x2= r2cos2&theta;
    y2= r2sin2&theta;

    What is x2+ y2?

    If that's too complicated, what is (x,y) when &theta;= 0?
    What is (x,y) when &theta;= &pi;?
     
  6. Nov 15, 2003 #5
    tq. u helped me solved the problem.

    but there is another part of the question which i ahain need some idea.

    --> prove that the real part of (1/z) is (1/2) for all values of [the]
     
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