Euler's Formula/Real Domain/Proof 1

Theorem

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

where $e^\cdot$ is the complex exponential function, $\cos$ is cosine, $\sin$ is sine, and $i$ is the imaginary unit.

Thus we define the complex exponential function in terms of standard trigonometric functions.

Direct Proof 1
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.

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

Step 3
Consider the initial condition $f\left({0}\right) = 1$.

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

But a specific solution to a differential equation is unique.

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