# Euler's Formula/Real Domain/Proof 1

## Contents

## Theorem

Let $\theta \in \R$ be a real number.

Then:

- $e^{i \theta} = \cos \theta + i \sin \theta$

## Proof

Consider the differential equation:

- $D_z f\left({z}\right) = i \cdot f\left({z}\right)$

### Step 1

We will prove that $z = \cos \theta + i \sin \theta$ is a solution.

\(\displaystyle z\) | \(=\) | \(\displaystyle \cos \theta + i \sin \theta\) | |||||||||||

\(\displaystyle \frac {\mathrm dz}{\mathrm d\theta}\) | \(=\) | \(\displaystyle -\sin \theta + i\cos \theta\) | Derivative of Sine Function, Derivative of Cosine Function, Linear Combination of Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle i^2\sin \theta + i\cos \theta\) | $i^2 = -1$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle i\left(i\sin\theta + \cos \theta\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle iz\) |

$\Box$

### Step 2

We will prove that $y = e^{i\theta}$ is a solution.

\(\displaystyle y\) | \(=\) | \(\displaystyle e^{i\theta}\) | |||||||||||

\(\displaystyle \frac {\mathrm dy}{\mathrm d\theta}\) | \(=\) | \(\displaystyle ie^{i\theta}\) | Derivative of Exponential Function, Chain Rule, Linear Combination of Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle iy\) |

$\Box$

### Step 3

Consider the initial condition $f \left({0}\right) = 1$.

\(\displaystyle \left.{y}\right \vert_{\theta \mathop = 0}\) | \(=\) | \(\displaystyle e^{0i}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \left.{z}\right \vert_{\theta \mathop = 0}\) | \(=\) | \(\displaystyle \cos 0 + i \sin 0\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) |

So $y$ and $z$ are both specific solutions.

But a specific solution to a differential equation is unique.

Therefore $y = z$, that is, $e^{i \theta} = \cos \theta + i \sin \theta$.

$\blacksquare$