We are asked to write the complex number z=-1-i in polar form.
The polar form of a complex number is
`z=r(cos theta + i sin theta)`
where r is the distance from the origin (the modulus or absolute value of the complex number) and theta is the angle from the positive x-axis. (Note that the angle is not unique -- you can always select an angle from 0 to 2pi or 0 to 360 degrees.)
To compute r we use
where a is the real part and b is the imaginary part of the complex number. So:
We can find theta by
`tan theta = b/a`
In this case we can solve the angle by inspection as 225 degrees or 5pi/4.
The polar form is:
An alternative is to write in Euler notation:
`z=|z|e^(i theta)=re^(i theta)`
So here we have:
Note again that the angle is not unique; we can add/subtract any multiples of 2pi and still have the same point. A general solution is:
`-1-i=sqrt(2)(cos((5pi)/4 + 2pi n)+isin((5pi)/4+ 2pi n))=sqrt(2)e^(i((5pi)/4+2pi n)); n in ZZ`