Euler's Formula

I know that Euler's identity is widely regarded as the most beautiful theorem in mathematics. In my opinion, the truly beautiful concept involved here is Euler's formula: \[ e^{ix} = \cos x + i \sin x. \] It unifies algebra, trigonometry, complex numbers, and calculus. Euler's identity is only a special case of Euler's formula, i.e., Euler's formula with \( x = \pi \) gives us Euler's identity: \[ e^{i \pi} = -1. \] This is cute but Euler's formula is truly beautiful. In fact with \( x = \tau = 2\pi, \) we get another cute result: \[ e^{i \tau} = 1. \] Quoting an excerpt from Chapter 22 of The Feynman Lectures on Physics, Volume I:

We summarise with this, the most remarkable formula in mathematics: \[ e^{i \theta} = \cos \theta + i \sin \theta. \] This is our jewel.

We may relate the geometry to the algebra by representing complex numbers in a plane; the horizontal position of a point is \( x, \) the vertical position of a point is \( y. \) We represent every complex number, \( x + iy. \) Then if the radial distance to this point is called \( r \) and the angle is called \( \theta, \) the algebraic law is that \( x + iy \) is written in the form \( r, e^{i \theta} \) where the geometrical relationships between \( x, \) \( y, \) \( r, \) and \( \theta \) are as shown. This, then, is the unification of algebra and geometry.

See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.