I'm having some trouble understanding this type of transformation. The materials provided by my professor doesn't even mention the method that is being used to switch from complex to sinusoidal and vise versa. For instance
I= -10(1 +j sqrt(3)/3)) .
Becomes,
i= 20/3 sqrt(6) sin(wt +210).
I only understand that 180 degrees are added because of the minus in the first part of the equation but what about the rest ?
here is another example in the opposite matter i(t) = 6 sin (wt + 3pi/4).
becomes..
I= -3+3j
The complex number ($I$ in your example) encodes the amplitude and the phase offset of the actual sinusoidal signal. For example, your first $I$ $$I = -10(1+j\sqrt3/3)$$ This is a number in rectangular form. Writing it in polar form,
$$I = Ae^{j\theta} = A\left(\cos\theta + j\sin\theta\right)\\ A\cos\theta = -10,\quad A\sin\theta = -\frac{10\sqrt3}{3}\\ A = \frac{20}{\sqrt3}\\ \tan\theta = \frac1{\sqrt3},\quad\theta = \frac{7\pi}{6}\\ I = \frac{20}{\sqrt3} e^{j7\pi/6} $$
The actual sinusoidal signal as a function of time is $$i(t) = \Re\left[\frac{20}{\sqrt3} e^{j(\color{red}{\omega t} + 7\pi/6)}\right]\\ = \frac{20}{\sqrt3} \cos\left(\color{red}{\omega t} + \frac{7\pi}6\right)$$