The polar form of a complex number is another way to represent a complex number. The form z = a + b i is called the rectangular coordinate form of a complex number. The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r and θ where r is the length of the vector and θ is the angle made with the real axis.

From Pythagorean Theorem :

r 2 = a 2 + b 2

By using the basic trigonometric ratios :

cos θ = a r and sin θ = b r .

Multiplying each side by r :

r cos θ = a and r sin θ = b

The rectangular form of a complex number is given by

z = a + b i .

Substitute the values of a and b .

z = a + b i         = r cos θ + ( r sin θ ) i         = r ( cos θ + i sin θ )

In the case of a complex number, r represents the absolute value or modulus and the angle θ is called the argument of the complex number.

This can be summarized as follows:

The polar form of a complex number z = a + b i is z = r ( cos θ + i sin θ ) , where r = | z | = a 2 + b 2 , a = r cos θ and b = r sin θ , and θ = tan − 1 ( b a ) for a > 0 and θ = tan − 1 ( b a ) + π or θ = tan − 1 ( b a ) + 180 ° for a 0 .

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Example:

Express the complex number in polar form.

5 + 2 i

The polar form of a complex number z = a + b i is z = r ( cos θ + i sin θ ) .

So, first find the absolute value of r .

r = | z | = a 2 + b 2         = 5 2 + 2 2         = 25 + 4         = 29           ≈ 5.39

Now find the argument θ .

Since a > 0 , use the formula θ = tan − 1 ( b a ) .

θ = tan − 1 ( 2 5 )         ≈ 0.38

Note that here θ is measured in radians.

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Therefore, the polar form of 5 + 2 i is about 5.39 ( cos ( 0.38 ) + i sin ( 0.38 ) ) .