Question

Express the following complex number on polar form : 1 + i​

Answer 1

$\large\underline{\sf{Solution-}}$

Given complex number is

$\rm \: 1 + i$

Let assume that

$\rm \: 1 + i = r(cos\theta + i \: sin\theta ) - - - (1)$

$\rm \: 1 + i = r \: cos\theta + i \:r \: sin\theta - - -$

So, on comparing real and Imaginary parts, we get

$\rm \: r \: cos\theta = 1 - - - - (2)$

$\rm \: r \: sin\theta = 1 - - - - (3)$

On squaring equation (2) and (3) and add, we get

$\rm \: {r}^{2} {cos}^{2}\theta + {r}^{2} {sin}^{2}\theta = 1 + 1$

$\rm \: {r}^{2}( {cos}^{2}\theta + {sin}^{2}\theta) = 2$

$\rm \: {r}^{2} = 2$

$\rm\implies \:r \: = \: \sqrt{2} - - - - (4)$

On substituting the value of r in equation (2) and (3), we get

$\rm \: cos\theta = \dfrac{1}{ \sqrt{2} }$

and

$\rm \: sin\theta = \dfrac{1}{ \sqrt{2} }$

$\rm \: As \: cos\theta \: and \: sin\theta \: are \: both \: positive.$

$\rm\implies \:\theta \: lies \: in \: {1}^{st} \: quadrant.$

$\rm\implies \:\theta = \dfrac{\pi}{4}$

So, required polar form is

$\rm\implies \:\boxed{\sf{  \: \: \rm \: 1 + i = \sqrt{2}\bigg(cos\dfrac{\pi}{4} + i \: sin \dfrac{\pi}{4} \bigg) \: \: }}$

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Additional Information :-

Argument of complex number :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Complex \: number & \bf arg(z) \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf x + iy & \sf  {tan}^{ - 1}\bigg |\dfrac{y}{x} \bigg|   \\ \\ \sf  - x + iy & \sf \pi - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf  - x - iy & \sf  - \pi + {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf x - iy & \sf  - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \end{array}} \\ \end{gathered}$

Answer 2

Step-by-step explanation:

very simple and easy method...

$\leadsto \tt1+i=r(\cos \theta+i \sin \theta) \Rightarrow \quad r \cos \theta=1, r \sin \theta=1$

$\text{\( \tt \Rightarrow \tan \theta=1, \: \: \: \theta=\dfrac{\pi}{4} \) and \( \tt r=\sqrt{2}\) }$

$\text{ \( \therefore \quad \) Polar form \( \red{ \tt =\sqrt{2}\left(\cos \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right) }\)}$