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

into the polar form and represent them in Argand plane.

Accepted Answer

Let $z = (1 - i)$

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 $r \sin θ = - 1$

Squaring and adding the both

$\Rightarrow r^{2} \cos^{2} θ + r^{2} \sin^{2} θ = 1^{2} + 1^{2}$

$\Rightarrow r^{2} = 2$

$∴ r = \sqrt{2} \to (1)$

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

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

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

$\Rightarrow θ = - \tan^{- 1} (1)$

$∴ θ = - \frac{π}{4} \to (2)$

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

The polar form of $(1 - i)$ is $\sqrt{2} (\cos (- \frac{π}{4}) + i \sin (- \frac{π}{4}))$ .

The polar form can be represented in Argand plane.

Here, $O P = \sqrt{2}$ and $∠ X O P = - \frac{π}{4}$ .

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