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Thought-Process to Discover Knowledge

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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)`. polar form of complex number `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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