Important Concepts and Formulas - Complex Numbers

Basics

A complex number is any number which can be written as $a + ib$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$

$a$ is the real part of the complex number and $b$ is the imaginary part of the complex number.

Example for a complex number: 9 + i2

$i^2 = -1$

$i^3 = -i$

$i^4 = 1$

if $z=a+ib$ is a complex number, a is called the real part of z and b is called the imaginary part of z.

It can be represented as Re(z) = a and Im(z) = b

Conjugate of the complex number $z=x+iy$ can be defined as $\bar{z} = x - iy$

Example: $\overline{4 + i2} = 4 - i2$ and $\overline{4 - i2} = 4 + 2i$

if the complex number $a + ib = 0$, then $a = b = 0$

if the complex number $a + ib = x + iy$, then $a = x$ and $b = y$

if $x + iy$ is a complex numer, then the non-negative real number $\sqrt{x^2 + y^2}$ is the modulus (or absolute value or magnitude) of the complex number $x + iy$. It can be denoted as

$\left| \ {x + iy} \ \right|=\sqrt{x^2 + y^2}$ (Note that modulus is a non-negative real number)

Rectangular(Cartesian) Form of Complex Numbers

A complex number when written in the form $a + ib$, it is in the Rectangular(Cartesian) form

$e^{i \theta} =\cos \theta+i\sin \theta $ (Euler's Formula)

Cube Roots of Unity = $(1)^{1/3} $

$= 1, \ \dfrac{-1 + i\sqrt{3}}{2}, \ \dfrac{-1 - i\sqrt{3}}{2}$

$= 1, \ w, \ w^2 $ where $w =\dfrac{-1 + i\sqrt{3}}{2}$

Properties of Cube Roots of Unity

(1) Cube Roots of Unity are in G.P.

(2) Each complex cube root of unity is the square of the other complex cube root of unity.

Example: $w =\dfrac{-1 + \sqrt{3}i}{2}, \ w^2 =\dfrac{-1 - \sqrt{3}i}{2}$

(3) $1 + w + w^2 = 0$

(4) Product of all cube roots of unity = 1
i.e., $w^3 = 1$

(5) $\dfrac{1}{w} = w^2$ and $\dfrac{1}{w^2} = w$

Fourth Roots of Unity , $(1)^{1/4}$ are +1, -1, +i, -i

Polar and Exponential Forms of Complex Numbers

Polar and Exponential Forms are very useful in dealing with the multiplication, division, power etc. of complex numbers.

Polar Form of a Complex Number

Polar Form of a Complex Number

A complex number $z=x+iy$ can be expressed in polar form as

$z=r \angle \theta = r \ \text{cis} \theta = r(\cos \theta+i\sin \theta) $ (Please not that θ can be in degrees or radians)

where $r =\left|z\right|=\sqrt{x^2 + y^2}$ (note that r ≥ 0 and and r = modulus or absolute value or magnitude of the complex number)

$\theta = \text{arg }z = \tan^{-1}{\left(\dfrac{y}{x}\right)}$(θ denotes the angle measured counterclockwise from the positive real axis.)

θ is called the argument of z. it should be noted that $2\pi \ n \ +\theta $ is also an argument of z where $n = \cdots -3, -2, -1, 0, 1, 2, 3, \cdots$. Note that while there can be many values for the argument, we will normally select the smallest positive value.

Please note that we need to make sure that θ is in the correct quadrant. i.e., θ should be in the same quadrant where the complex number is located in the complex plane. This will be clear from the next topic where we will go through various examples to convert complex numbers between polar form and rectangular form. It is strongly recommended to go through those examples to get the concept clear.

$x = r \ \cos \theta $
$y = r \ \sin \theta$

If $-\pi < \theta \leq\pi, \quad \theta$ is called as principal argument of z(In this statement, θ is expressed in radian)

Exponential form of a Complex Number

We have already seen that in polar form , a complex number can be expressed as $z=r(\cos \theta+i\sin \theta) $. By Euler's Formula, we have $e^{i \theta} =\cos \theta+i\sin \theta $

Hence, we can express a complex number in Exponential form as $z=re^{i \theta}$ (Note that θ is in radians)

While there may be many values of θ satisfying this, we will normally select the smallest positive value.

Note that radians and degrees are two units for measuring angles.

$360° = 2\pi$ radian

Convert Complex Numbers from Rectangular Form to Polar Form and Exponential Form

Example 1: Convert $z = 1 + i\sqrt{3}$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{1^2 + (\sqrt{3})^2}\\=\sqrt{1 + 3}=\sqrt{4} = 2$

$\text{arg }z =\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)} = \tan^{-1}{\left(\dfrac{\sqrt{3}}{1}\right)}\\= \tan^{-1}{\left(\sqrt{3}\right)} =\dfrac{\pi}{3}$

Here the complex number is in first quadrant in the complex plane. The angle we got, $\dfrac{\pi}{3}$ is also in the first quadrant. Hence we select this value.

Hence, the polar form is $z = 2 \angle{\left(\dfrac{\pi}{3}\right)} = 2\left[\cos\left(\dfrac{\pi}{3}\right)+i\sin\left(\dfrac{\pi}{3}\right)\right] $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 2e^{\left(\dfrac{i\pi}{3}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{\pi}{3} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 2: Convert $z = -1 + i\sqrt{3}$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(-1)^2 + (\sqrt{3})^2}\\=\sqrt{1 + 3}=\sqrt{4} = 2$

$\text{arg }z =\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)} = \tan^{-1}{\left(\dfrac{\sqrt{3}}{-1}\right)}\\= \tan^{-1}{\left(-\sqrt{3}\right)}$

Here $-\dfrac{\pi}{3}$ is one value of θ which meets the condition $\theta = \tan^{-1}{\left(-\sqrt{3}\right)}$. But it is in fourth quadrant. We know that θ should be in second quadrant because the complex number is in second quadrant in the complex plane.

Hence $θ = -\dfrac{\pi}{3}+\pi=\dfrac{2\pi}{3} $ which is in second quadrant and also meets the condition $\theta = \tan^{-1}{\left(-\sqrt{3}\right)}$. Hence we take that value.

Hence, the polar form is
$z = 2 \angle{\left(\dfrac{2\pi}{3}\right)} $ $= 2\left[\cos\left(\dfrac{2\pi}{3}\right)+i\sin\left(\dfrac{2\pi}{3}\right)\right] $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 2e^{\left(\dfrac{i \ 2\pi}{3}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{2\pi}{3} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 3: Convert $z = -1 - i\sqrt{3}$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}\\=\sqrt{(-1)^2 + (-\sqrt{3})^2}\\=\sqrt{1 + 3}=\sqrt{4} = 2$

$\text{arg }z =\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)}\\= \tan^{-1}{\left(\dfrac{-\sqrt{3}}{-1}\right)}=\tan^{-1}{\left(\sqrt{3}\right)}$

Here $\dfrac{\pi}{3}$ is one value of θ which meets the condition $\theta = \tan^{-1}{\left(\sqrt{3}\right)}$. But it is in first quadrant. We know that θ should be in third quadrant because the complex number is in third quadrant in the complex plane.

Hence $θ =\dfrac{\pi}{3}+\pi=\dfrac{4\pi}{3} $ which is in third quadrant and also meets the condition $\theta = \tan^{-1}{\left(\sqrt{3}\right)}$. Hence we take that value.

Hence, the polar form is
$z = 2 \angle{\left(\dfrac{4\pi}{3}\right)} $ $= 2\left[\cos\left(\dfrac{4\pi}{3}\right)+i\sin\left(\dfrac{4\pi}{3}\right)\right] $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 2e^{\left(\dfrac{i \ 4\pi}{3}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{4\pi}{3} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 4: Convert $z = 1 - i\sqrt{3}$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(1)^2 + (-\sqrt{3})^2}\\=\sqrt{1 + 3}=\sqrt{4} = 2$

$\text{arg }z =\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)}\\= \tan^{-1}{\left(\dfrac{-\sqrt{3}}{1}\right)}\\= \tan^{-1}{\left(-\sqrt{3}\right)}$

Here $-\dfrac{\pi}{3}$ is one value of θ which meets the condition and also in the fourth quadrant. The complex number is also in fourth quadrant.However we will normally select the smallest positive value for θ.

Hence $\theta = -\dfrac{\pi}{3}+2\pi=\dfrac{5\pi}{3}$ which meets the condition $\theta = \tan^{-1}{\left(\sqrt{3}\right)}$ and also is in the fourth quadrant. Hence we take that value.

Hence, the polar form is
$z = 2 \angle{\left(\dfrac{5\pi}{3}\right)}$ $= 2\left[\cos\left(\dfrac{5\pi}{3}\right)+i\sin\left(\dfrac{5\pi}{3}\right)\right] $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 2e^{\left(\dfrac{i \ 5\pi}{3}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{5\pi}{3} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 5: Convert $z = 8$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(8)^2 + (0)^2}\\=\sqrt{(8)^2 } = 8$

Here the complex number lies in the positive real axis. Hence $\theta = 0$.

$\text{arg }z =\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)} = \tan^{-1}{\left(\dfrac{0}{8}\right)}\\= \tan^{-1}{0}=0$

Hence, the polar form is $z = 8 \angle{0} = 8\left(\cos 0+i\sin 0\right) $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 8e^{0i}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + 0 = 2\pi n$ where $n=0, \pm 1, \pm 2, \cdots$ Accordingly we can get other possible polar forms and exponential forms also)

Example 6: Convert $z = -8$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(-8)^2 + (0)^2}\\=\sqrt{(-8)^2 } = 8$

Here the complex number lies in the negative real axis. Hence $\theta =\pi$.

Hence, the polar form is $z = 8 \angle{\pi} = 8\left(\cos\pi+i\sin\pi\right) $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 8e^{i\pi}$

(Please note that all possible values of the argument, arg z are $2\pi n+\pi \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 7: Convert $z = i8$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(0)^2 + (8)^2}\\=\sqrt{(8)^2 } = 8$

Here the complex number lies in the positive imaginary axis. Hence $\theta =\dfrac{\pi}{2}$

Hence, the polar form is
$z = 8 \angle{\dfrac{\pi}{2}}=8\left[\cos\left(\dfrac{\pi}{2}\right)+i\sin\left(\dfrac{\pi}{2}\right)\right]$

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 8e^{\left(\dfrac{i\pi}{2}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{\pi}{2} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Example 8: Convert $z = -i8$ to polar form

$r =\left|z\right|=\sqrt{x^2 + y^2}=\sqrt{(0)^2 + (-8)^2}\\=\sqrt{(-8)^2 } = 8$

Here the complex number lies in the negavive imaginary axis. Hence $\theta = -\dfrac{\pi}{2}$.

However we will normally select the smallest positive value for θ. Hence $\theta = -\dfrac{\pi}{2}+2\pi=\dfrac{3\pi}{2}$

Hence, the polar form is
$z = 8 \angle{\dfrac{3\pi}{2}}$ $=8\left[\cos\left(\dfrac{3\pi}{2}\right)+i\sin\left(\dfrac{3\pi}{2}\right)\right] $

Similarly we can write the complex number in exponential form as $z=re^{i \theta} = 8e^{\left(\dfrac{i 3\pi}{2}\right)}$

(Please note that all possible values of the argument, arg z are $2\pi \ n \ + \dfrac{3\pi}{2} \text{ where } n = 0, \pm 1, \pm 2, \cdots$. Accordingly we can get other possible polar forms and exponential forms also)

Convert Complex Numbers from Polar Form to Rectangular(Cartesian) Form

Example 1: Convert $2\left[\cos\left(\dfrac{5\pi}{3}\right)+i\sin\left(\dfrac{5\pi}{3}\right)\right]$ to Rectangular(Cartesian) form

$x=r\cos\theta$ $= 2 \cos \dfrac{5\pi}{3} = 2 \times \dfrac{1}{2} = 1$

$y=r\sin\theta$ $= 2 \sin \dfrac{5\pi}{3} = 2 \times\left(-\dfrac{\sqrt{3}}{2}\right) = -\sqrt{3}$

$z=x+iy = 1 - i\sqrt{3}$

Example 2: Convert $8\left[\cos\left(\dfrac{\pi}{2}\right)+i\sin\left(\dfrac{\pi}{2}\right)\right]$ to Rectangular(Cartesian) form

$x=r\cos\theta= 8 \cos \dfrac{\pi}{2} = 2 \times 0 = 0$

$y=r\sin\theta= 8 \sin \dfrac{\pi}{2} = 8 \times 1 = 8$

$z=x+iy = 0 + 8i = 8i$

Convert Complex Numbers from Exponential Form to Rectangular(Cartesian) Form

Example 1: Convert $2e^{\left(\dfrac{i2\pi}{3}\right)}$ to Rectangular(Cartesian) form

$x=r\cos\theta$ $= 2 \cos \dfrac{2\pi}{3} = 2 \times\left(-\dfrac{1}{2}\right)= -1$

$y=r\sin\theta$ $= 2 \sin \dfrac{2\pi}{3} = 2 \times \dfrac{\sqrt{3}}{2}=\sqrt{3}$

$z=x+iy = -1 + i\sqrt{3}$

Example 2: Convert $2e^{\left(\dfrac{i\pi}{3}\right)}$ to Rectangular(Cartesian) form

$x=r\cos\theta= 2 \cos \dfrac{\pi}{3} = 2 \times \dfrac{1}{2}= 1$

$y=r\sin\theta= 2 \sin \dfrac{\pi}{3} = 2 \times \dfrac{\sqrt{3}}{2}=\sqrt{3} $

$z=x+iy = 1 + i\sqrt{3}$

Arithmetical Operations of Complex Numbers

Addition, subtraction, multiplication and division can be carried out on complex numbers in either rectangular form or polar form.

Addition and subtraction of complex numbers is easy in rectangular form.

Multiplication and division of complex numbers is easy in polar form.

Addition of Complex Numbers

To add complex numbers, add their real parts and add their imaginary parts.

(a + ib) + (c + id) = (a + c) + i(b + d)

Example: Find (9 + i2) + (8 + i6)

(9 + i2) + (8 + i6) = (9 + 8) + i(2 + 6) = 17 + i8

Subtraction of Complex Numbers

To subtract complex numbers, subtract their real parts and subtract their imaginary parts.

(a + ib) - (c + id) = (a - c) + i(b - d)

Example: Find (9 + 2i) - (8 + 6i)

(9 + 2i) - (8 + 6i) = (9 - 8) + i(2 - 6) = 1 - i4

Multiplication of Complex Numbers

A. Multiplication of Complex Numbers in Rectangular Form

(a + ib)(c + id) = ac + iad + ibc + i²bd = ac + iad + ibc - bd = (ac - bd) + i(ad + bc)

Example: Find (9 + 2i)(8 - 6i)

$(9 + i2)(8 - i6)\\= 72 - i54 + i16 - i^2 12\\= 72 - i(54 - 16) + 12\\= 84 - i38$

B. Multiplication of Complex Numbers in Polar Form

$r_1 \angle \theta_1 \times r_2 \angle \theta_2 = r_1 r_2 \angle\left(\theta_1 + \theta_2\right)$

Example: Find $3\angle 30° \times 4\angle 40°$

$3\angle 30° \times 4\angle 40°\\=\left(3 \times 4\right) \angle\left(30° + 40°\right)\\= 12 \angle 70°$

Division of Complex Numbers

A. Division of Complex Numbers in Rectangular Form

$\dfrac{(a + ib)}{(c + id)}\\~\\=\dfrac{(a + ib)}{(c + id)} \times \dfrac{(c - id)}{(c - id)}\\~\\=\dfrac{(ac + bd) - i(ad - bc)}{c^2 + d^2}$

(We multiplied denominator and numerator with the conjugate of the denominator to proceed)

Example: Find $\dfrac{(9 + 2i)}{(8 - 6i)}$

$\dfrac{(9 + 2i)}{(8 - 6i)}\\~\\=\dfrac{(9 + 2i)(8 + 6i)}{(8 - 6i)(8 + 6i)}\\~\\=\dfrac{72 + 54i + 16i -12}{64 + 36}\\~\\=\dfrac{60 + 70i}{100}\\ = .6 + .7i$

B. Division of Complex Numbers in Polar Form

$\dfrac{r_1 \angle \theta_1}{r_2 \angle \theta_2} =\dfrac{r_1}{r_2} \angle\left(\theta_1 - \theta_2\right)$

Example: Find $\dfrac{5\angle 135° }{4\angle 75°}$

$\dfrac{5\angle 135° }{4\angle 75°} =\dfrac{5}{4}\angle\left( 135° - 75°\right) =\dfrac{5}{4}\angle 60° $

Powers of Complex Numbers

De'Moivre's Theorem

If n is any integer, then $(\cos \theta+i\sin \theta)^n =\cos n \theta+i\sin n \theta$

From De'Moivre's formula, it is clear that for any complex number

$z=r \text{ cis }\theta=r \angle \theta$ $=r(\cos \theta+i\sin \theta) = r e^{i \theta}$

$z^n=r^n \angle\left(\theta \times n\right)$ $=r^n(\cos n \theta+i\sin n \theta)$ $= r^n e^{in \theta}$

Example 1: Find $\left(-1 + \sqrt{3} \ i\right)^{12}$

$r=\sqrt{\left(-1\right)^2 +\left(\sqrt{3}\right)^2}\\=\sqrt{1 + 3}=\sqrt{4} = 2$

$\theta = \tan^{-1}{\left(\dfrac{\sqrt{3}}{-1}\right)} = \tan^{-1}{\left(-\sqrt{3}\right)}\\=\dfrac{2\pi}{3}$
(∵The complex number is in second quadrant)

Hence,

$-1 + \sqrt{3} \ i\\= 2\left[\cos\left(\dfrac{2\pi}{3}\right)+i\sin\left(\dfrac{2\pi}{3}\right)\right]$

$\Rightarrow\left(-1 + \sqrt{3} \ i\right)^{12}\\=\left[2\left(\cos \ \dfrac{2\pi}{3}+i\sin \ \dfrac{2\pi}{3}\right)\right]^{12}\\= 2^{12}\left[ \cos\left(\dfrac{2\pi}{3} \times 12\right)+i\sin\left( \dfrac{2\pi}{3} \times 12\right)\right]\\= 4096 \left( \cos 8\pi +i\sin 8\pi\right)\\= 4096 \left( \cos 0 +i\sin 0\right)\\= 4096(1 + 0) = 4096$

Example 2: Find $\left(2 \angle 135°\right)^5$

$\left(2 \angle 135°\right)^5 = 2^5\left(\angle 135° \times 5\right)\\= 32 \angle 675° = 32 \angle -45°\\=32\left[\cos (-45°)+i\sin (-45°)\right]\\=32\left[\cos (45°) - i\sin (45°)\right]\\= 32\left(\dfrac{1}{\sqrt{2}}-i \dfrac{1}{\sqrt{2}}\right)\\=\dfrac{32}{\sqrt{2}}(1-i)$

Example 3: Find $\left[4\left(\cos 30°+i\sin 30°\right)\right]^6$

$\left[4\left(\cos 30°+i\sin 30°\right)\right]^6 \\= 4^6\left[\cos\left(30° \times 6\right)+i\sin\left(30° \times 6\right)\right]\\=4096\left(\cos 180°+i\sin 180°\right)\\=4096(-1+i\times 0)\\=4096 \times (-1)\\=-4096$

Example 4: Find $\left(2e^{0.3i}\right)^8$

$\left(2e^{0.3i}\right)^8 = 2^8e^{\left(0.3i \times 8\right)} = 256e^{2.4i}\\=256(\cos 2.4+i\sin 2.4)$

Roots of Complex Numbers

n^th roots of a complex number $z=r(\cos \theta+i\sin \theta)$ can be given by

$w_k$ $=r^{1/n}\left[\cos\left(\dfrac{\theta + 2\pi k }{n}\right)+i\sin\left(\dfrac{\theta + 2\pi k}{n}\right)\right]$

where k = 0, 1, 2, . . . (n-1)

(If θ is in degrees, substitute 360° for $2\pi$)

Example 1: Find 5^th roots of 32i

$32i = 32\left(\cos \dfrac{\pi}{2}+i\sin \dfrac{\pi}{2}\right)\quad$ (converted to polar form, reference)

The 5^th roots of 32i can be given by

$w_k\\=r^{1/n}\left[\cos\left(\dfrac{\theta + 2\pi k}{n}\right)+i\sin\left(\dfrac{\theta + 2\pi k}{n}\right)\right]\\=32^{1/5}\left[\cos\left(\dfrac{\dfrac{\pi}{2}+2\pi k }{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+2\pi k}{5}\right)\right]\\=2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+2\pi k }{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+2\pi k}{5}\right)\right]$

where k = 0, 1, 2,3 and 4

$w_0 = 2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+0}{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+0}{5}\right)\right]$ $= 2\left(\cos \dfrac{\pi}{10}+i\sin \dfrac{\pi}{10}\right)$

$w_1 = 2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+2\pi}{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+2\pi}{5}\right)\right]$ $= 2\left(\cos \dfrac{\pi}{2}+i\sin \dfrac{\pi}{2}\right) = 2i$

$w_2 = 2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+4\pi}{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+4\pi}{5}\right)\right]$ $= 2\left(\cos \dfrac{9\pi}{10}+i\sin \dfrac{9\pi}{10}\right)$

$w_3 = 2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+6\pi}{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+6\pi}{5}\right)\right]$ $= 2\left(\cos \dfrac{13\pi}{10}+i\sin \dfrac{13\pi}{10}\right)$

$w_4 = 2\left[\cos\left(\dfrac{\dfrac{\pi}{2}+8\pi}{5}\right)+i\sin\left(\dfrac{\dfrac{\pi}{2}+8\pi}{5}\right)\right]$ $= 2\left(\cos \dfrac{17\pi}{10}+i\sin \dfrac{17\pi}{10}\right)$

Example 2: Find cube roots of $-4 - 4\sqrt{3}i$

$-4 - 4\sqrt{3}i = 8\left(\cos 240°+i\sin 240°\right)\quad$
(converted to polar form, reference. Here we took the angle in degrees. Remember that we can use radians or degrees)

The cube roots of $-4 - 4\sqrt{3}i$ can be given by

$w_k\\=r^{1/n}\left[\cos\left(\dfrac{\theta + 360°k }{n}\right)+i\sin\left(\dfrac{\theta + 360°k}{n}\right)\right]\\=8^{1/3}\left[\cos\left(\dfrac{\text{240°+360°k}}{3}\right)+i\sin\left(\dfrac{\text{240°+360°k}}{3}\right)\right]\\=2\left[\cos\left(\dfrac{240°+ 360°k }{3}\right)+i\sin\left(\dfrac{240° + 360°k}{3}\right)\right]$
where k = 0, 1 and 2

$w_0\\=2\left[\cos\left(\dfrac{240°+ 0}{3}\right)+i\sin\left(\dfrac{240° + 0}{3}\right)\right]\\= 2\left(\cos 80°+i\sin 80°\right)$

$w_1\\=2\left[\cos\left(\dfrac{\text{240°+360°}}{3}\right)+i\sin\left(\dfrac{\text{240°+360°}}{3}\right)\right]\\=2\left(\cos 200°+i\sin 200°\right)$

$w_2\\=2\left[\cos\left(\dfrac{240°+ 720°}{3}\right)+i\sin\left(\dfrac{240° + 720°}{3}\right)\right]\\ =2\left(\cos 320°+i\sin 320°\right)$

Example 3: Find cube roots of 1

$1=1\left(\cos 0+i\sin 0\right)$(Converted to polar form, reference. Here we took the angle in degrees. Remember that we can use radians or degrees)

The cube roots of 1 can be given by

$w_k\\=r^{1/n}\left[\cos\left(\dfrac{\theta + 360°k }{n}\right)+i\sin\left(\dfrac{\theta + 360°k}{n}\right)\right]\\\\=1^{1/3}\left[\cos\left(\dfrac{\text{0°+360°k}}{3}\right)+i\sin\left(\dfrac{\text{0°+360°k}}{3}\right)\right]\\=\cos (120°k)+i\sin (120°k)$

where k = 0, 1 and 2

$w_0 =\cos\left(120° \times 0\right)+i\sin\left(120°\times 0\right)$ $=\cos 0+i\sin 0 = 1$

$w_1 =\cos\left(120° \times 1\right)+i\sin\left(120°\times 1\right)\\=\cos 120°+i\sin 120°\\=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\\=\dfrac{-1 + i\sqrt{3}}{2}$

$w_2 =\cos\left(120° \times 2\right)+i\sin\left(120°\times 2\right)\\=\cos 240°+i\sin 240°\\=-\dfrac{1}{2} - i\dfrac{\sqrt{3}}{2}\\=\dfrac{-1 - i\sqrt{3}}{2}$

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