Question

Express in polar form: (-1-i)​

Answer 1

Given:

The complex number is

`-1-i`

To find:

The polar form of given complex number.

Solution:

If z=a+ib, then

`r=|z|=√(a^2+b^2)`

`\tan \theta = (b)/(a)`

The polar form is
`z=r(\cos \theta +i\sin \theta)`.

Let the complex number is

`z=-1-i`

Here, a=-1, b=-1. Both a and b are negative. So, z lies in III quadrant.

`r=√((-1)^2+(-1)^2)`

`r=√(1+1)`

`r=√(2)`

And,

`\tan \theta = (-1)/(-1)`

`\tan \theta = 1`

`\tan \theta = \tan (\pi+(\pi)/(4))` (because z lies in III quadrant)

`\tan \theta = \tan ((5\pi)/(4))`

`\theta = (5\pi)/(4)`

Now, the polar form of given complex number is

`z=r(\cos \theta +i\sin \theta)`

`z=√(2)(\cos (5\pi)/(4) +i\sin (5\pi)/(4))`

Therefore, the required polar form is
`z=√(2)(\cos (5\pi)/(4) +i\sin (5\pi)/(4))`.