Euler's Formula/Real Domain/Proof 1

Proof
Consider the differential equation:


 * $D_z \map f z = i \cdot \map f z$

Step 1
We will prove that $z = \cos \theta + i \sin \theta$ is a solution.

Step 2
We will prove that $y = e^{i\theta}$ is a solution.

Step 3
Consider the initial condition $\map f 0 = 1$.

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

But a particular solution to a differential equation is unique.

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