# Polar form of a complex number solutions

Polar form of a complex number. If z= x + iy is a complex number. Then z = r ( cosθ + sinθ ) is called polar form or trigonometric form of a complex number.

By comparing real and imaginary parts of a complex number,  we get
x = r cosθ           (1)
y = r sinθ           (2)

By squaring above equations and adding we get
$r^2&space;=&space;x^2&space;+y^2$

EXAMPLE

Express $-\sqrt{3}&space;-i$ in polar form.

Solution

With $x&space;=-\sqrt{3}$ and y = −1 we obtain

$r&space;=&space;|z|&space;=\sqrt{(\sqrt{-3})^2+&space;(-1)^2}=&space;2$
Now

$\frac{y}{x}&space;=&space;\frac{-1}{-\sqrt{3}}=\frac{1}{\sqrt{3}}\$
and
$\theta=tan^{-1}(\frac{y}{x})=\frac{1}{\sqrt{3}}=&space;\frac{\pi}{6}\$

which is an angle whose terminal side is in the first quadrant.But

since the point $(-\sqrt{3},&space;-1)$ lies in the third quadrant, we take the solution
$tan&space;\theta&space;=\frac{-1}{\sqrt{-3}}=\frac{1}{\sqrt{3}}$
So,
$\theta&space;=&space;arg(z)&space;=&space;\frac{\pi}{6}+&space;\pi&space;=&space;\frac{7\pi}{6}$
Therefore, polar form of the number is
$z&space;=&space;2(cos\frac{7\pi}{6}&space;+&space;isin&space;\frac{7\pi}{6})$

Solved Questions of polar form
• Polar of  $\frac{-34i}{5-3i}$  click here
• Polar for of  $(-2+2i)(1-i)$     click here
• Polar for of  $-\sqrt{3}+&space;i$  click here
• Polar for of  $-i$      click here
• Polar for of  $-1-\sqrt{3}i$   click here
• Polar for of    $-1+i$      click here
CHALLENGE

write the given complex number in polar form first using an argument θ = Arg(z)
1.       2
2.    −10
3.     −3i
4.       6i
5.       1 + i
6.       5 − 5i
7.    $-\sqrt{3}+&space;i$ click here for solution
8.    $-2&space;-&space;2\sqrt{3i}$
9.    $\frac{3}{-1&space;+&space;i}$
10.  $\frac{12}{\sqrt{3}&space;+&space;i}$

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1. Plz provide the solution