Question

Express the complex number in trigonometric form.

3 - 3i

Answer 1

Factoring a 3sqrt2 out of each term gives us:\[\Large\rm 3\sqrt2\left(\frac{\sqrt2}{2}-\frac{\sqrt2}{2}\mathcal i\right)\]

With the square root 2's we're at one of our pi/4 values right?
Cosine is giving us a positive value, sine negative, so we're in the 4th quadrant.

So if we want the principle angle from -pi to pi, our angle would be -pi/4\[\Large\rm 3\sqrt2\left(\cos\left[-\frac{\pi}{4}\right]+\mathcal i \sin\left[-\frac{\pi}{4}\right]\right)\]

Answer 2

If that first step was too confusing, another approach is to remember,\[\Large\rm \theta=\arctan\left(\frac{y}{x}\right)\]

Answer 3

So you would have:\[\Large\rm \theta=\arctan\left(\frac{-3}{3}\right)\]