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.