# Operations on complex numbers in polar form

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##### Examples
###### Lessons
1. Multiplying complex numbers in polar form
1. $4(\cos(\frac{5\pi}{3})+i \sin(\frac{5\pi}{3})) \cdot 8(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$
2. $(\cos(170^{\circ})+i \sin(170^{\circ}))\cdot 5(\cos(45^{\circ})+i \sin(45^{\circ}))$
3. $3(\cos(\pi)+i \sin(\pi))\cdot(\cos(\frac{\pi}{5})+i \sin(\frac{\pi}{5}))\cdot6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$
2. Dividing complex numbers in polar form
1. $20(\cos(\frac{5\pi}{2})+i \sin(\frac{5\pi}{2}))\div 6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$
2. $3(\cos(\frac{3\pi}{4})+i \sin(\frac{3\pi}{4}))\div 12(\cos(\frac{\pi}{6})+i \sin(\frac{\pi}{6}))$
3. $(\cos(262^{\circ})+i \sin(262^{\circ}))\div (\cos(56^{\circ})+i \sin(56^{\circ}))$
3. Convert the following complex number to exponential form
$z=3+i$
###### Topic Notes
Let's find out how to perform some basic operations on complex numbers in polar form! We will briefly introduce the notion of the exponential form of a complex number, then we will focus on multiplication and division on complex numbers in polar form.
Note:

Polar form
real part
$a=|z|\cos \theta$
imaginary part
$b=|z|\sin \theta$
$z=|z|(\cos \theta+i\sin \theta)$

When …
Multiplying: multiply the absolute values, and add the angles
Dividing: divide the absolute values, and subtract the angles

Exponential form
$z=|z|e^{i \theta}$