Find the amplitude (the angle) of this complex number
$$z= 2\sqrt{2}\,i - 2\sqrt{3}$$
I am getting that the modulus is $2\sqrt{5}$. After that I am getting $\cos -\sqrt{3}/{\sqrt{5}}$ and $\sin\sqrt{2}/\sqrt{5}$. How to find the answer?
See also my previous question, How to find the amplitude of a complex number $z=-1-\sqrt{3}i$.
The angle you seek is usually called the argument of a complex number.
You have correctly found the modulus. The complex number $z = 2\sqrt{2}i - 2\sqrt{3} = 2\sqrt{3} - 2\sqrt{2}i$ is represented by the point $(-2\sqrt{3}, 2\sqrt{2})$ of the complex plane, as shown below.
Since the angle $\theta$ is in the second quadrant and \begin{align*} \arcsin x: & [-1, 1] \to \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\\ \arccos x: & [-1, 1] \to [0, \pi]\\ \arctan x: & (-\infty, \infty) \to \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \end{align*} we can most easily find the angle by finding the arccosine of the angle. Hence, $$\theta = \arccos\left(\frac{-2\sqrt{3}}{2\sqrt{5}}\right) = \arccos\left(-\frac{\sqrt{3}}{\sqrt{5}}\right)$$
Note: Cosine and Sine take an angle as their argument. You should have written \begin{align*} \cos\theta & = -\frac{\sqrt{3}}{\sqrt{5}}\\ \sin\theta & = \frac{\sqrt{2}}{\sqrt{5}} \end{align*}