2

When writing a complex number, which is polar form? $re^{i \theta}$ or $r(\cos \theta + i \sin \theta)$?

Googling will give different answers, some websites call the first exponential form and the other for polar form. Others call the first polar form and the second trigonometric form.

Which is correct? Are both polar form maybe?

RadiansForLife
  • 133
  • http://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-eit-cos-t-i-sin-t – lab bhattacharjee Mar 02 '17 at 10:45
  • 3
    Both formulas give the same complex number, so they are identical. You can call both "polar form". Its up to your preference (or that of the author of whatever book/lecture you are reading). – Simon Mar 02 '17 at 10:52
  • the first form is exponential form and the second is polar form – David Quinn Mar 02 '17 at 10:57
  • 1
    @DavidQuinn The first is the exponential form of the polar form and the second is the trigonometric form of the polar form. The "polar" originates from the parameters (argument and absolute value) but not from the usage of specific functions. – Reinhard Meier Mar 02 '17 at 11:03

1 Answers1

2

Euler's famous formula says

$$e^{i\theta}=\cos(\theta)+i\sin(\theta).$$

Then

$$re^{i\theta}=r(\cos(\theta)+i\sin(\theta)).$$

You will sometimes find the notations

$$r\text{ cis}(\theta)$$ or $$r\angle\theta.$$

You can indeed call these polar forms, reminding the identities

$$\begin{cases}x=r\cos(\theta),\\y=r\sin(\theta).\end{cases}$$