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  4. If z=r( cos θ +i sin θ ), then the value of (z/z)=( overlinez/z)

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Q. If $ z=r(\cos \theta +i\sin \theta ), $ then the value of $ \frac{z}{z}=\frac{\overline{z}}{z} $

KEAMKEAM 2010Complex Numbers and Quadratic Equations

Solution:

We have $ z=r(\cos \theta +i\sin \theta ) $
$ \therefore $ $ \overline{z}=r(\cos \theta -i\sin \theta ) $
$ \therefore $ $ \frac{z}{z}+\frac{\overline{z}}{z} $
$=\frac{r(\cos \theta +i\sin \theta )}{r(\cos \theta -i\sin \theta )}+\frac{r(\cos \theta -i\sin \theta )}{r(\cos \theta +i\sin \theta )} $
$=\frac{{{(\cos \theta +i\sin \theta )}^{2}}+(\cos \theta -i\sin {{\theta }^{2}})}{{{\cos }^{2}}\theta +{{\sin }^{2}}\theta } $
$={{\cos }^{2}}\theta -{{\sin }^{2}}\theta +2i\cos \theta \sin \theta +{{\cos }^{2}}\theta $ $ -{{\sin }^{2}}\theta +2i\cos \theta \sin \theta $
$=2({{\cos }^{2}}\theta -{{\sin }^{2}}\theta ) $
$=2\cos 2\theta $