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Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers and Quadratic Equations2 Marks
Question
Represent the complex number z = 1 + $i \sqrt { 3 }$ in the polar form.
✓

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
r2 (cos2$\theta$ + sin2$\theta$) = 1 + 3
$\Rightarrow$ r2 = 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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