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mathsComplex NumbersComplex Plane and Polar Form

### Polar form of a Complex Number

Complex numbers, ordered pair of real numbers, are alternatively given by modulus and argument. This is the polar form of the complex number.

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While introducing the generic form of complex numbers a+i b, it was discussed an equivalent form r(cos theta + i sin theta). A quick revision of the same with complex plane is given here.

A number a+b i is equivalently given as r(cos theta + i sin theta) where r = sqrt(a^2+b^2) and theta = tan^(-1)(b/a). r(cos theta + i sin theta) is called the polar form or polar representation of the complex number.

Polar form of a complex number is r (cos theta + i sin theta)

Polar Form: A number in the form a+bi is equivalently given as
quad quad = r (cos (theta+ 2n pi)
quad quad quad quad + i sin (theta+ 2n pi))
where r = sqrt(a^2+b^2),
theta = tan^(-1)(b/a).

What is the form of complex number given by r (cos theta + sin theta)?

• Pronunciation : Say the answer once
Spelling: Write the answer once

The answer is 'Polar form'.

Solved Exercise Problem:

Convert 1+i into polar form.

• 2(cos(pi/4)+i sin(pi/4))
• sqrt(2)(cos(pi/4)+i sin(pi/4))
• sqrt(2)(cos(pi/4)+i sin(pi/4))
• cos(pi/4)+i sin(pi/4)

The answer is 'sqrt(2)(cos(pi/4)+i sin(pi/4))'

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