The basic idea is that to any sinusoidal quantity \$x(t)\$ with angular frequency \$\omega\$ you can associate a constant complex quantity \$X\$ so that $$x(t) = \operatorname{Re}[X\mathrm{e}^{\mathrm{j}\omega t}],\qquad\qquad(1)$$ where \$\operatorname{Re}\$ denotes the real part of the complex number \$X\mathrm{e}^{\mathrm{j}\omega t}\$. The quantity \$X\$ is called the phasor associated to \$x(t)\$. This kind of representation is also called Steinmentz transform.
Other representations are possible, e.g, the associated phasor can be chosen so that $$x(t) = \operatorname{Re}[\sqrt{2}\,X\mathrm{e}^{\mathrm{j}\omega t}],\qquad\qquad(2)$$ in this way the modulus \$|X|\$ of the complex number \$X\$ yields the root-mean-square (RMS) value of \$x(t)\$. Or one can choose the phasor so that \$x(t)\$ is represented by the imaginary parts $$x(t) = \operatorname{Im}[X\mathrm{e}^{\mathrm{j}\omega t}],\qquad\qquad(3)$$ or $$x(t) = \operatorname{Im}[\sqrt{2}\,X\mathrm{e}^{\mathrm{j}\omega t}].\qquad\qquad(4)$$
In the case of your examples, with a bit of "reverse engineering", it's easy to see that your professor chose the representation (4).
Let's consider your last example with \$I = (-3+3\mathrm{j})\,\mathrm{A}\$ (complex quantities have units too!). The polar representation of \$I\$ is $$I = (-3+3\mathrm{j})\,\mathrm{A} = 3\sqrt{2}\mathrm{e}^{\mathrm{j}\frac{3\pi}{4}}\,\mathrm{A}.$$ Thus, $$i(t) = \operatorname{Im}[\sqrt{2}\,I\mathrm{e}^{\mathrm{j}\omega t}] = \operatorname{Im}\left[6\mathrm{e}^{\mathrm{j}\left(\omega t+\frac{3\pi}{4}\right)}\,\mathrm{A}\right],$$ and by using the Euler's formula $$\mathrm{e}^{\mathrm{j}\alpha} = \cos\alpha+\mathrm{j}\sin\alpha,$$ we obtain
$$\begin{align}i(t) &= \operatorname{Im}\left\{6\left[\cos\left(\omega t+\frac{3\pi}{4}\right)+\mathrm{j}\sin\left(\omega t+\frac{3\pi}{4}\right)\right]\,\mathrm{A}\right\}, \\ &= 6\sin\left(\omega t+\frac{3\pi}{4}\right)\,\mathrm{A}.\end{align}$$
