Operations on complex numbers in polar form

Operations on complex numbers in polar form

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.

Lessons

Note:

Polar form
real part
a=zcosθa=|z|\cos \theta
imaginary part
b=zsinθb=|z|\sin \theta
z=z(cosθ+isinθ)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=zeiθz=|z|e^{i \theta}
  • 1.
    Multiplying complex numbers in polar form
    a)
    4(cos(5π3)+isin(5π3))8(cos(2π3)+isin(2π3)) 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}))

    b)
    (cos(170)+isin(170))5(cos(45)+isin(45)) (\cos(170^{\circ})+i \sin(170^{\circ}))\cdot 5(\cos(45^{\circ})+i \sin(45^{\circ}))

    c)
    3(cos(π)+isin(π))(cos(π5)+isin(π5))6(cos(2π3)+isin(2π3))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
    a)
    20(cos(5π2)+isin(5π2))÷6(cos(2π3)+isin(2π3)) 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}))

    b)
    3(cos(3π4)+isin(3π4))÷12(cos(π6)+isin(π6)) 3(\cos(\frac{3\pi}{4})+i \sin(\frac{3\pi}{4}))\div 12(\cos(\frac{\pi}{6})+i \sin(\frac{\pi}{6}))

    c)
    (cos(262)+isin(262))÷(cos(56)+isin(56)) (\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+iz=3+i