The Proof
Now we have the tools to prove why \(e^{i\theta} = \cos\theta + i \sin\theta\). We simply take the Maclaurin series for \(e^x\) and replace \(x\) with \(i\theta\).
\[e^{i\theta} = 1 + (i\theta) + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \dots\]
Using the cycle of \(i\) (\(i^2=-1, i^3=-i, i^4=1\)):
\[e^{i\theta} = (1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots) + i(\theta - \frac{\theta^3}{3!} + \dots)\]
The first part is exactly the Cosine series, and the second part is exactly the Sine series!
Worked Examples
Example 1: The Magic of Taylor
This proof shows that sines and cosines are not just geometric shapes; they are the fundamental components of the exponential function when rotation is involved. This is why every wave in the universe can be described by an exponent.
The Bridge to Quantum Mechanics
In Quantum Mechanics, we represent the "Momentum" of a particle as a rotation in the complex plane: \(\psi = e^{ikx}\). According to Euler's formula, this means a particle with a definite momentum is a perfect mixture of a cosine wave (the real part) and a sine wave (the imaginary part). Without this power series proof, we wouldn't understand why momentum and position are related by a Fourier Transform. This identity is the "glue" that holds the dual nature of light and matter together.