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

1 - i \sqrt{3}

into the polar form and represent them in Argand plane.

Accepted Answer

Let $z = (1 - i \sqrt{3})$

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

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

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

$\Rightarrow r^{2} = 4$

$∴ r = 2 \to (1)$

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

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

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

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

$∴ θ = - \frac{π}{3} \to (2)$

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

The polar form of $(1 - i \sqrt{3})$ is $2 (\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3}))$ .

The polar form can be represented in Argand plane.

Here $O P = 2$ and $∠ X O P = - \frac{π}{3}$ .

Convert the complex number $1 - i \sqrt{3}$ into the polar form and represent them in Argand plane.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Convert the complex number 1-i3 into the polar form and represent them in Argand plane.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Convert the complex number $1 - i \sqrt{3}$ into the polar form and represent them in Argand plane.