Question

Express the complex number √2+4i in the polar form.

Answer 1

Given :-

Complex number is $\sqrt{2}+4i$

To find :-

Polar form of complex number

Solution :-

In order to represent any complex number in polar form, firstly we need to calculate it's modulus and argument.

Let us assume,

• $y = Im(z)=4$
• $x = Re(z) =\sqrt{2}$

• Modulus of a complex number is given by,

${ \implies |z| = r = \sqrt{ {\sf{Re}}(z)^2+{ \sf{Im}}(z)^2} }$

${ \implies |z| = r = \sqrt{ ( \sqrt{2} )^2+(4)^2} }$

${ \implies |z| = r = \sqrt{ 2+16 }}$

${ \implies |z| = r = \sqrt{ 18}}$

${ \implies |z| = r = 3\sqrt{2}}$

So the modulus of given complex number is $3\sqrt{2}$.

• Argument of a complex number in first quadrant is given by,

$\leadsto \theta = \tan^ - \left| \dfrac{Im(z)}{Re(z)}\right|$

$\leadsto \theta = \tan^ - \left| \dfrac{y}{x}\right|$

$\leadsto \theta = \tan^ - \left| \dfrac{4}{ \sqrt{2} }\right|$

${ \leadsto \theta = \tan^ - \left| \dfrac{ \sqrt{2} \times \sqrt{2} \times \sqrt{2} \times \sqrt{2} }{ \sqrt{2} }\right| }$

${ \leadsto \theta = \tan^ - \left| 2 \sqrt{2} \right| }$

So this is the argument of the complex number.

• Every complex number in polar form can be expressed as,

$\dashrightarrow \: \: r \bigg( \cos \theta + i \sin \theta\bigg)$

Substitute the value of r and θ.

${ \dashrightarrow \: \: 3 \sqrt{2} \bigg( \cos( \tan^ - \left| 2 \sqrt{2} \right|) + i \sin( \tan^ - \left| 2 \sqrt{2} \right| )\bigg)}$

This is the required polar form.

‎ ‎ ‎

It can also be written in short as,

$\longrightarrow \underline { \underline{3 \sqrt{2} cis(\tan^ - \left| 2 \sqrt{2} \right| )}}$

Additional Information :-

Argument of complex number in 1st quadrant is given by $\theta$ where $\theta$ is measured anti-clockwise.

Argument of complex number in 2nd quadrant is given by $\pi - \theta$ where $\theta$ is measured anti-clockwise.

Argument of complex number in 3rd quadrant is given by $\theta-\pi$ where $\theta$ is measured clockwise .

Argument of complex number in 4th quadrant is given by $-\theta$ where $\theta$ is measured clockwise.

Answer 2

Answer:

Given Equation:

• Z = √2 + 4i

Comparing it with the general equation of z = a + bi, we get:

• a = √2
• b = 4

To convert a complex number to the polar form we first find |z| which is called the argument of the number. The argument can be calculated by the formula:

⇒ |z| = √ (a² + b²)

⇒ |z| = √ ((√2)² + 4²)

⇒ |z| = √ (2 + 16)

⇒ |z| = √18 = 3√2

Now we need to calculate the value of θ. This can be calculated by using the formula,

$\boxed{ \bf{ \theta = tan^{-1}[\dfrac{b}{a}]}}$

Hence the value of θ is:

$\implies \theta = arctan(\:\:\dfrac{4}{\sqrt{2}}\:\:)\\\\\\\implies \theta = arctan(\:\:\dfrac{4\sqrt{2}}{2}\:\:)\\\\\\\implies \theta = arctan(\:\:2\sqrt{2}\:\:)\\\\\\\implies \boxed{\bf{\theta = 70.5^\circ} \:\:or\:\:1.23\:\;rad}$

Hence the polar form of (a + bi) is given as:

⇒ Polar Form = |z| ( cosθ + i sinθ )

⇒ Polar Form = 3√2 ( cos (1.23) + i sin(1.23) )