Name: Geometric View of Complex Number
Uploaded: 2019-07-19
Duration: 10 min 19 s
Description: Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis. Watch the video to l...

Question

Convert the complex number

\frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3 i}}

into polar form.

Accepted Answer

Let $z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3 i}}$

$\Rightarrow z = \frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3} i} \times \frac{5 - \sqrt{3} i}{5 - \sqrt{3} i}$ (Multiply and divide by conjugate of denominator)

$\Rightarrow z = \frac{10 - 2 \sqrt{3} i + 30 \sqrt{3} i - 18 i^{2}}{25 - 3 i^{2}}$

$\Rightarrow z = \frac{10 + 28 \sqrt{3} i + 18}{25 + 3} [∵ i^{2} = - 1]$

$\Rightarrow z = \frac{28 + 28 \sqrt{3} i}{28}$

$\Rightarrow z = 1 + \sqrt{3} i$

We know the polar form of $x + i y$ is $z = r (\cos θ + i \sin θ) .$

Here $r$ is magnitude of $z$ and $θ$ is argument of $z$ .

On comparing with given complex form , we get

$r \cos θ = 1$ and $r \sin θ = \sqrt{3}$

Squaring and adding the both,we get

$\Rightarrow r^{2} \cos^{2} θ + r^{2} \sin^{2} θ = 1^{2} + {(\sqrt{3})}^{2}$

$\Rightarrow r^{2} = 1 + 3$

$\Rightarrow r^{2} = 4$

$∴ r = 2 \to (1)$

$∴ \cos θ = \frac{1}{2} and \sin θ = \frac{\sqrt{3}}{2}$

$\Rightarrow θ = \tan^{- 1} (\frac{\sin θ}{\cos θ})$

$\Rightarrow θ = \tan^{- 1} (\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}})$ (Since $x, y$ lies in first quadrant)

$\Rightarrow θ = \tan^{- 1} (\sqrt{3})$

$∴ θ = \frac{π}{3} \to (2)$

From $(1)$ and $(2)$ ,

The polar form of $\frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3 i}}$ is $2 (\cos \frac{π}{3} + i \sin \frac{π}{3})$ .

Convert the complex number $\frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3 i}}$ into polar form.

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Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Convert the complex number $\frac{2 + 6 \sqrt{3} i}{5 + \sqrt{3 i}}$ into polar form.