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Math Help - Complex Numbers - Find z & w

  1. #1
    Newbie Flamuri's Avatar
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    Red face Complex Numbers - Find Z & W

    Assign Z & W [ Complex numbers ] if

    z + w = 1 - i ;

    argZ = \frac{\pi}{6} ;

    argW = \frac{5\pi }{3}

    Have tried so many times, but no results, can someone please help me explaining those kind of tasks

    Thanks,
    Flamuri
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  2. #2
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    Re: Complex Numbers - Find Z & W

    Well, we can see that \displaystyle z = r_1\left(\cos{\frac{\pi}{6}} + i\sin{\frac{\pi}{6}}\right) = r_1\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = \frac{\sqrt{3}}{2}r_1 + \frac{1}{2}ir_1 and \displaystyle w = r_2\left(\cos{\frac{5\pi}{3}} + i\sin{\frac{5\pi}{3}}\right) = r_2\left(\frac{1}{2} - \frac{\sqrt{3}}{2}i\right) = \frac{1}{2}r_2 - \frac{\sqrt{3}}{2}ir_2.

    So \displaystyle z + w = \frac{\sqrt{3}}{2}r_1 + \frac{1}{2}r_2 + \left(\frac{1}{2}r_1 - \frac{\sqrt{3}}{2}r_2\right)i.

    Now equate your real and imaginary parts to give you two equations to solve for \displaystyle r_1 and \displaystyle r_2 and you will have \displaystyle z and \displaystyle w.
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Complex Numbers - Find Z & W

    modified.
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  4. #4
    Newbie Flamuri's Avatar
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    Re: Complex Numbers - Find Z & W

    Wow, I've used a completely different way [for me a bit harder way] from this that you have used, but finally found a result:

    If

    z = a + bi
    w = x + yi

    So:

    z = - \frac{\sqrt{3}+3}{2} + \frac{\sqrt{3}-1}{2}i

    w = \frac{\sqrt{3}+3}{2} + \frac{\sqrt{3}+1}{2}i

    Don't know how to prove if i'm right, maybe you can help me again,
    Thanks.
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    Re: Complex Numbers - Find Z & W

    Quote Originally Posted by Flamuri View Post
    Assign Z & W [ Complex numbers ] if
    z + w = 1 - i ;

    argZ = \frac{\pi}{6} ; argW = \frac{5\pi }{3}
    I would do it this way. Let z=a+bi~\&~w=c+di.

    From the given \begin{align*}  a+c&= 1 \\  b+d &=-1 \end{align*} and \frac{b}{a} = \tan \left( {\frac{\pi }{6}} \right)\;\& \,\frac{d}{c} = \tan \left( {\frac{{5\pi }}{3}} \right)
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  6. #6
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    Re: Complex Numbers - Find Z & W

    Quote Originally Posted by Flamuri View Post
    Wow, I've used a completely different way [for me a bit harder way] from this that you have used, but finally found a result:

    If

    z = a + bi
    w = x + yi

    So:

    z = - \frac{\sqrt{3}+3}{2} + \frac{\sqrt{3}-1}{2}i

    w = \frac{\sqrt{3}+3}{2} + \frac{\sqrt{3}+1}{2}i

    Don't know how to prove if i'm right, maybe you can help me again,
    Thanks.
    I don't know what you've done here... You're told \displaystyle z + w = 1 - i and we've found

    \displaystyle z + w = \frac{\sqrt{3}}{2}r_1 + \frac{1}{2}r_2 + \left(\frac{1}{2}r_1 - \frac{\sqrt{3}}{2}r_2\right)i.

    Surely that means

    \displaystyle \frac{\sqrt{3}}{2}r_1 + \frac{1}{2}r_2 = 1 and \displaystyle \frac{1}{2}r_1 - \frac{\sqrt{3}}{2}r_2 = -1.

    Solve these equations simultaneously for \displaystyle r_1 and \displaystyle r_2.
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