H2 Math Question on Complex Numbers

• LastRide7

  25 Jan 12, 01:22

  The complex number q is given by q = (e^iθ)/(1 - e^iθ) where 0<θ<2pi. Find the real part of q.

   

  I don't know how to begin the question.

  
  

• frekiwang

  25 Jan 12, 06:25

  One way to do it is just by a direct approach, knowing e^iθ=cosθ+isinθ:

  q

  =(cosθ+isinθ)/(1-cosθ-isinθ)

  =[(cosθ+isinθ)(1-cosθ+isinθ)]/[(1-cosθ-isinθ)(1-cosθ+isinθ)]

  =[(cosθ+isinθ-(cosθ)^2-isinθcosθ+isinθcosθ-sin^θ]/[(1-cosθ)^2+(sinθ)^2]

  =(cosθ+isinθ-1)/[1-2cosθ+(cosθ)^2+(sinθ)^2]

  =(cosθ-1)/(2-2cosθ)+i[sinθ/(2-2cosθ)]

  therefore, the real part is (cosθ-1)/(2-2cosθ)=(cosθ-1)/2(1-cosθ)=-1/2

  There is a short-cut to this, I will write after school if you need it.

• wee_ws

  25 Jan 12, 07:16

  Hi,

    We try to make the denominator of q real, which is done by multiplying q top and bottom by e^(-iθ/2) giving

    e^(iθ/2) / [ e^(-iθ/2) - e^(iθ/2) ].

    Since e^(iθ/2) + e^(-iθ/2) = 2 cos (θ/2), the above expression simplifies to

    [ cos (θ/2) + isin (θ/2) ] / [ -2 cos (θ/2) ], so that

    real(q) = -1/2.

    I have written about this type of problem before, so you may find it useful to read it:

    http://sgforums.com/forums/2297/topics/356377?page=3#post_10226565

    Thanks.

  Cheers,
  Wen Shih