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?