Question

How do I find the trigonometric form of the complex number `3-3sqrt3 i`?

Answer 1

In trigonometric form: `6(cos 5.236+i sin 5.236)`

Explanation:

Let `Z=a+ib ; Z=3- 3 sqrt 3i ; a=3 ,b = - 3sqrt3` ;

`Z` is in `4` th quadrant. Modulus `|Z|=sqrt(a^2+b^2)`

`=(sqrt(3^2+ (-3 sqrt3)^2)) =sqrt 36 =6`

`tan alpha =|b/a|= (3sqrt3)/3 or tan alpha =sqrt 3`

`alpha = tan ^-1 (sqrt3) ~~ 1.0472`

`theta` is on `4`th quadrant `:. theta=2pi-1.0472~~ 5.236:.`

Argument , `theta =5.236 :.` In trigonometric form expressed as

`|Z|(cos theta+i sin theta) = 6(cos 5.236+i sin 5.236)` [Ans]