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$ e $ and Trigonometry: Euler's Formula

The Taylor series of $ e^x $ is

                $ \begin{align} e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots \end{align} $

Using this equation, it is possible to relate $ e $ to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, $ ix $ for example. This yields:

                $ \begin{align} e^{ix} &= \sum^{\infty}_{n=0}{\frac{(ix)^n}{n!}}\\ &= \sum^{\infty}_{n=0}{\frac{i^nx^n}{n!}}\\ &= 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \cdots\\ &= (1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots) + i(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots)\\ &= \sum^{\infty}_{n=0}{\frac{(-1^nx^{2n}}{(2n)!}} + i\sum^{\infty}_{n=0}{\frac{(-1)^nx^{2n+1}}{(2n+1)!}}\\ &= \cos(x) + i\sin(x) \end{align} $

This surprising result was first published by Euler in 1748. It leads to a large number of significant insights regarding all of its parts. The first important corollary is that every nonzero complex number can be written as $ e^{a+bi} $. This follows from $ \{\cos(x)+i\sin(x) | x \in \mathbb{R}\} $ being the unit circle: it is obvious that every point can be written as a unit circle point multiplied by a scalar.

This leads to an insight regarding complex multiplication: First, observe that $ \left|e^{a+bi}\right| = e^a $. This indicates that the length and angle of the complex number are directly separated when the number is written as $ e $ to a complex number. Let $ z_1 = e^{a_1+b_1i} $ and $ z_2 = e^{a_2+b_2i} $. Then $ z_1\cdot z_2 = e^{a_1+b_1i}\cdot e^{a_2+b_2i} = e^{a_1+a_2} \cdot e^{i(b_1+b_2)} $.

Conceptually, this means that the length of the two points are multiplied to get the length of the new point. A similar thing applies to the points' angles: they are added rather than multiplied. This is an important idea fundamental to intuitively understanding the multiplication of complex numbers.

Many trigonometric identities become clear and nearly trivial when Euler's formula is applied to them. For example:

                $ \begin{align} e^{i\theta} &= \cos(\theta) + isin(\theta)\\ e^{i\theta}\cdot e^{-e\theta} &= (\cos(\theta) + isin(\theta))(\cos(\theta) - isin(\theta))\\ e^{i\theta-i\theta} &= \cos^2(\theta) - i^2\sin^2(\theta)\\ 1 &= \cos^2\theta + \sin^2\theta \end{align} $

Euler's formula also provides alternative definitions of sine and cosine:

                $ \begin{align} \cos(\theta) = \frac1{2}\left(e^{i\theta}+e^{-i\theta}\right)\\ \sin(\theta) = \frac1{2i}\left(e^{i\theta}-e^{-i\theta}\right) \end{align} $

References:
(Reference 1)
(Reference 2)