Question

How do you write the trigonometric form of `-2-2i`?

Answer 1

The trigonometric form is `z=2sqrt2(cos((5pi)/4)+isin((5pi)/4))`

Explanation:

Let `z=-2-2i`
The modulus of z is `|z|=sqrt(2^2+2^2)=2sqrt2`
Then we rewrite z as `z=2sqrt2(-2/(2sqrt2)-2/(2sqrt2)i)`
simplify `z=2sqrt2(-sqrt2/2-sqrt2/2i)`
Comparing this to `z=r(costheta+isintheta)` which is the trgonometric form.
So `costheta=-sqrt2/2` and `sintheta=-sqrt2/2`
`:. theta` is in the 3rd quadrant
`theta=(5pi)/4`
So the trigonometric form is #z=2sqrt2(cos((5pi)/4)+isin((5pi)/4))

and the exponential form is `z=2sqrt2e^((5ipi)/4)`