Euler's Formula/Real Domain/Proof 2

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 2
This:


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

is logically equivalent to this:


 * $\displaystyle \dfrac{\cos \theta + i \sin \theta} {e^{i \theta} } = 1$

for every $\theta$.

Note that the left expression is nowhere undefined.

Taking the derivative of this:

Thus the expression, as a function of $\theta$, is constant and so yields the same value for every $\theta$.

We know the value at at least one point, that is, when $\theta = 0$:


 * $\displaystyle \dfrac{\cos 0 + i \sin 0}{e^{0 i}} = \dfrac {1 + 0} 1 = 1$

Thus it is $1$ for every $\theta$, which verifies the above.

Hence the result.