After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the modulus of the complex number plus the cosine (real part) and the sine (imaginary part) of the angle found by the inverse of the tangent function, while the Euler form works the same but without the modulus stated. Am I right?
PS. If, for instance, I have to express $z = 1 + i$ in Euler form, I just find the inverse of the tangent function: $1/1 = 1$. I ask myself the question: what are the angles for which the tangent is $1$? $\pi / 4$ and $5\pi/4$ (but I exclude the second angle because drawing the complex number in rectangular form, I know that it lies on the first quadrant) and I write finally: $z = cos(\pi/4), i sin(\pi/4)$. Am I right?
I was taught "Euler's form" in engineering mechanics and it was for the 3D case (implicitly quaternions..without the 1/2 scale factor you'd have if directly using the similarity transform) instead of the 2D case (complex numbers). But since the function is analytic it doesn't really matter, we are taking the principle valued log of the standard representation where the principle value is taken to be the torque minimal (minimum magnitude) implied angle. So we have:
$\theta \in \left(-\pi,\pi\right]$
The only "difference" then is the various ways you can define the principle valued logarithm.