Euler's formula links complex exponentials with trigonometric functions:
\[ e^{ix} = \cos x + i \sin x \]
\[ e^{i\pi} = \cos \pi + i \sin \pi = -1 + 0 = -1 \]
\[ \Rightarrow e^{i\pi} + 1 = 0 \]
\[ r = \sqrt{3^2 + 4^2} = 5 \]
\[ x = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.927 \]
\[ 3 + 4i = 5e^{0.927i} \]
\[ e^{z} = e^{x + iy} = e^x (\cos y + i \sin y) \]
Euler’s formula elegantly unifies exponential, trigonometric, and complex number concepts—revealing the deep structure of mathematics and natural phenomena.