Represent the complex number z = 1 + i 3 in the polar form.

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 $1 + i \sqrt { 3 } = r ( \cos \theta + i \sin \theta) ...(i)$
On equating real and imaginary parts both sides, we get
$r \cos \theta = 1$ and$ r \sin \theta = \sqrt { 3 } ...(ii)$
On squaring and adding Eqs. (i) and (ii), we get
$r^2 (\cos^2 \theta + \sin^2 \theta) = 1 + 3$
$\Rightarrow r^2 = 4$
$\Rightarrow r = 2$
$\therefore \cos \theta = \frac { 1 } { 2 }$ 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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