Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCOMPLEX NUMBERS AND QUADRATIC EQUATIONS2 Marks
Question
Represent the complex number z = 1 + $i \sqrt { 3 }$ in the polar form.
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Answer
We have, $z=1+i \sqrt{3}$
Let $1+i \sqrt{3}=r(\cos \theta+i \sin \theta)$,
On equating real and imaginary parts both sides, we get $r \cos \theta=1$ and $r \sin \theta=\sqrt{3} \ldots$...(ii)
On squaring and adding Eqs. (i) and (ii), we get
$r^2\left(\cos ^2 \theta+\sin ^2 \theta\right)=1+3$
$\Rightarrow r^2=4$
$\Rightarrow r=2$
$\therefore \cos \theta=\frac{1}{2}$ and $\sin \theta=\frac{\sqrt{3}}{2}$
Since, both $\cos \theta$ and $\sin \theta$ are positive.
So, $\theta$ lies in first quadrant.
$\therefore \theta=\frac{\pi}{3}$
On putting $r=2$ and $\theta=\frac{\pi}{3}$ in Eq. (i), we get polar form of $z=2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$
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