# Difference between Euler form and polar / trig form of a complex number

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?

• Consider that the readers may not know what you mean by "the Euler form":
– user147263
Commented Oct 24, 2015 at 22:08
• As I have already said, is not me who speaks of "Euler form" but is the prof. who wrote the exercise. This is why I have asked the question here. I have never heard before of "Euler form" too! Heard only about polar or trigonometric form. Commented Oct 25, 2015 at 4:27
• Then, after some readings yesterday, discovered about the Euler formula. Perhaps, improperly, the prof. wrote "Euler form" thinking abot the Euler formula. However, still haven't solved my doubt! Commented Oct 25, 2015 at 4:29
• "prof. who wrote the exercise. This is why I have asked the question here" This is a non sequitur, you should ask the prof. who wrote the exercise.
– user147263
Commented Oct 25, 2015 at 4:32
• I wanted to avoid it but I have asked to him already. I have asked here as, at the level of my knowledge, I did not think it was negligence of prof. Now, after your confirmation, I know it. Commented Oct 25, 2015 at 10:31

$\theta \in \left(-\pi,\pi\right]$