Question

Express the complex number in trigonometric form. -6 + 6 sqrt3 i

Answer 1

find the angle and the |-6+6sqrt()3|i

Answer 2

You can begin by finding the absolute value (distance to the origin) of the complex number: \(r = \sqrt{\mathbf{Re}^2 + \mathbf{Im}^2} = \sqrt{(-6)^2 + (6\sqrt3)^2} = \sqrt{36 + 108} = \sqrt{144} = 12)\).

Answer 3

If you divide your number by the absolute value, the absolute value of the new number will be 1 but the angle will be preserved (see figure). Call the new number \(s = (-6 + 6\sqrt3i)/12 = -\frac12 + \frac{\sqrt3}2i\). The real part is \(\mathbf{Re}(s) = -\frac12\) but also \(\cos\theta\) (the angle in the figure). Solving this equaion for theta we get \(\theta = 120^\circ\).