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Math Help - Polar Form C.N

  1. #1
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    Question Polar Form C.N

    If
    z = -8 sqrt(3) + 8 i
    AND
    w = (-(11/sqrt(2))/2) + ((11/sqrt(2))/2)i

    determine z / w in polar form:
    r cis(theta)

    Am stuck on the above, how do i solve it?
    Thanks
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  2. #2
    Member Danshader's Avatar
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    given :

    <br />
z = -8\sqrt{3} +8i<br />
    and
    <br />
w = <br />
\frac{-11}{2\sqrt{2}} + \frac{11}{2\sqrt{2}}i<br />

    now
    <br />
\frac{z}{w} = \frac{-8\sqrt{3} +8i}{<br />
\frac{-11}{2\sqrt{2}} + \frac{11}{2\sqrt{2}}i}<br />

    multiplying the numerator and the denominator with the conjugate of w we get:

    <br />
\frac{z}{w} = \frac{(-8\sqrt{3} +8i)( \frac{-11}{2\sqrt{2}} - \frac{11}{2\sqrt{2}}i)}{(<br />
\frac{-11}{2\sqrt{2}} + \frac{11}{2\sqrt{2}}i)( \frac{-11}{2\sqrt{2}} - \frac{11}{2\sqrt{2}}i)}<br />
    <br />
\frac{z}{w} = \frac{\frac{88\sqrt{3}}{2\sqrt{2}} +\frac{88\sqrt{3}}{2\sqrt{2}}i - \frac{88}{2\sqrt{2}}i +\frac{88}{2\sqrt{2}}}{\frac{121}{8} +\frac{121}{8}}<br />
    <br />
\frac{z}{w} = \frac{\frac{88}{2\sqrt{2}}(\sqrt{3}+1) +\frac{88}{2\sqrt{2}}(\sqrt{3}-1)i}{\frac{121}{4}}<br />
    <br />
\frac{z}{w} = \frac{16}{11\sqrt{2}}(\sqrt{3}+1) +\frac{16}{11\sqrt{2}}(\sqrt{3}-1)i<br />

    to get the r value of the polar form is actually the modulus of \frac{z}{w}

    \left| \frac{z}{w} \right| = \sqrt{ \left( \frac{16}{11\sqrt{2}}(\sqrt{3}+1)\right) ^2 +\left( \frac{16}{11\sqrt{2}}(\sqrt{3}-1)\right) ^2}<br />
    <br />
\left| \frac{z}{w} \right| = \frac{32}{11} = r<br />

    to find the angle \theta we use the fact that:

    if z = a +bi
    <br />
tan (\theta) = \frac{b}{a}<br />

    therefore in our case it should be:
    <br />
tan(\theta) = \frac{\frac{16}{11\sqrt{2}}(\sqrt{3}-1)}{\frac{16}{11\sqrt{2}}(\sqrt{3}+1)}<br />
    <br />
\theta =15^o <br />

    the polar representation of a complex number is in the form of:
    <br />
z = r(cos(\theta) + isin(\theta))<br />

    hence,
    <br />
\left| \frac{z}{w} \right| = \frac{32}{11}(cos(15^o) + isin(15^o))<br />
    <br />
\left| \frac{z}{w} \right|= \frac{32}{11}(cos(\frac{\pi}{12}) + isin(\frac{\pi}{12}))<br />
    <br />
\left| \frac{z}{w} \right|= \frac{32}{11}cis(\frac{\pi}{12})<br />

    hope i didn't mess things up.
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