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 Monday 20th of May

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 Depdendent Variable

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 Dependent Variable

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# Polar Form of a Complex Number

Example 1

Writing a Complex Number in Polar Form

Write the complex number in polar form.

Solution

The absolute value of z is

and the angle θ is given by

Because and because lies in Quadrant III, you choose θ to be θ = π + π/3 = 4π/3. Thus, the polar form is

(See the figure below)

The polar form adapts nicely to multiplication and division of complex numbers. Suppose you are given two complex numbers

z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 )

The product of z1 and z2 is

 z1 z2 = r1 r2 cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) = r1 r2 [(cos θ1cos θ2 - i sin θ1sin θ2) + i(sin θ1cos θ2 + cos θ1sin θ2)]

Using the sum and difference formulas for cosine and sine, you can rewrite this equation as

z1 z2 = r1 r2 [(cos1+ θ2) + i sin (θ1+ θ2)]

This establishes the first part of the following rule. Try to establish the second part on your own.

Product and Quotient of Two Complex Numbers

Let z1 = r1(cos θ1 + i sin θ1 ) and z2 = r2(cos θ2 + i sin θ2 ) be complex numbers.

 z1 z2 = r1 r2 [(cos (θ1+ θ2) + i sin (θ1+ θ2)] Product Quotient

Note that this rule says that to multiply two complex numbers you multiply moduli and add arguments, whereas to divide two complex numbers you divide moduli and subtract arguments.