Operations on complex numbers in polar form - Complex Numbers

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.

Basic concepts:
Exponents: Product rule $(a^x)(a^y)=a^{(x+y)}$
Exponents: Division rule ${a^x \over a^y}=a^{(x-y)}$

Related concepts:
Imaginary zeros of polynomials

Lessons

Notes:

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

1.
Multiplying complex numbers in polar form
2.
Dividing complex numbers in polar form

Operations on complex numbers in polar form

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AlgebraExponents: Product rule (ax)(ay)=a(x+y) (a^x)(a^y)=a^{(x+y)}(ax)(ay)=a(x+y)

AlgebraExponents: Division rule axay=a(x−y){a^x \over a^y}=a^{(x-y)}ayax​=a(x−y)

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Exponents: Product rule (ax)(ay)=a(x+y) (a^x)(a^y)=a^{(x+y)}(ax)(ay)=a(x+y)

Exponents: Division rule axay=a(x−y){a^x \over a^y}=a^{(x-y)}ayax​=a(x−y)

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Lesson: 1b

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Lesson: 2a

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Lessons

Notes:

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})) 4(cos(35π​)+isin(35π​))⋅8(cos(32π​)+isin(32π​))

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})) (cos(170∘)+isin(170∘))⋅5(cos(45∘)+isin(45∘))

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})) 3(cos(π)+isin(π))⋅(cos(5π​)+isin(5π​))⋅6(cos(32π​)+isin(32π​))

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})) 20(cos(25π​)+isin(25π​))÷6(cos(32π​)+isin(32π​))

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})) 3(cos(43π​)+isin(43π​))÷12(cos(6π​)+isin(6π​))

c)

(cos⁡(262∘)+isin⁡(262∘))÷(cos⁡(56∘)+isin⁡(56∘)) (\cos(262^{\circ})+i \sin(262^{\circ}))\div (\cos(56^{\circ})+i \sin(56^{\circ})) (cos(262∘)+isin(262∘))÷(cos(56∘)+isin(56∘))

3.

Convert the following complex number to exponential form z=3+iz=3+iz=3+i

Operations on complex numbers in polar form

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Exponents: Product rule $(a^x)(a^y)=a^{(x+y)}$

Algebra
Exponents: Division rule ${a^x \over a^y}=a^{(x-y)}$

Exponents: Product rule $(a^x)(a^y)=a^{(x+y)}$

Exponents: Division rule ${a^x \over a^y}=a^{(x-y)}$

or

$4(\cos(\frac{5\pi}{3})+i \sin(\frac{5\pi}{3})) \cdot 8(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$

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

$3(\cos(\pi)+i \sin(\pi))\cdot(\cos(\frac{\pi}{5})+i \sin(\frac{\pi}{5}))\cdot6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$

$20(\cos(\frac{5\pi}{2})+i \sin(\frac{5\pi}{2}))\div 6(\cos(\frac{2\pi}{3})+i \sin(\frac{2\pi}{3}))$

$3(\cos(\frac{3\pi}{4})+i \sin(\frac{3\pi}{4}))\div 12(\cos(\frac{\pi}{6})+i \sin(\frac{\pi}{6}))$

$(\cos(262^{\circ})+i \sin(262^{\circ}))\div (\cos(56^{\circ})+i \sin(56^{\circ}))$

Convert the following complex number to exponential form
$z=3+i$