Question

Convert the following complex number into its polar representation 2+2i

Answer 1

`z=2√(2)(cos{(\pi)/(4)}+isin{(\pi)/(4)})`

Explanation:

The given complex number is:

`2+2i`

Now, the polar from of the complex number
`z=a+ib` is
`z=r(cos{\theta}+isin{\theta})`.

Finding the absolute value of r:

`r=|z|={√(2^2+2^2)`

`r=\sqrt8`

`r=2√(2)`

Now, find the value of argument,

Using formula,
`{\theta}=tan^(-1)((b)/(a))`

`{\theta}=tan^(-1)({(2)/(2))`

`{\theta}=tan^(-1)(1)`

`{\theta}={(\pi)/(4)}`

Thus, the polar form is:

`z=2√(2)(cos{(\pi)/(4)}+isin{(\pi)/(4)})`

which is the required polar form.

Answer 2

To write 2+2i into polar form we proceed as follows:
2+2i
simplifying the above we get
2(1+i)
dividing through above through by √2
=2*√2(1/√2+i/√2)
but
1/√2=cos (π/4)=sin (π/4)
thus our expression will be:
=2√2(cos (π/4)+isin (π/4))