Operations on complex numbers in polar form

  1. Multiplying complex numbers in polar form
    1. 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}))
    2. (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}))
    3. 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
    1. 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}))
    2. 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}))
    3. (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
    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.

    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}