Question

How do you write the complex number `z = −2 − 2isqrt3` in trigonometric form?

Answer 1

As you can see this is the `30^0, 60^0, 90^0` right angle triangle which happen to have `2, sqrt(3), 1` relationship between the sides.
The phasor that you were given is scale by 2. See image...
From this the magnitude of you Phasor, P:`|P| = 4`
and the direction angle is `alpha = 240^0` see image
Thus you can write:
`P = 4cos240 + i4sin240 = -2 - i3.46 = -2 -i2sqrt(2)`

Explanation:

See above it is self evident on the image...