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Operations on complex numbers in polar form

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  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
    z=3+iz=3+i
    Topic Notes
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    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⁡θa=|z|\cos \theta
    imaginary part
    b=∣z∣sin⁡θb=|z|\sin \theta
    z=∣z∣(cos⁡θ+isin⁡θ)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∣eiθz=|z|e^{i \theta}
    Basic Concepts
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    Related Concepts
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