$ 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, "Euler's Formula", was first published by Euler in 1748. It leads to a large number of significant insights regarding all of its parts. It is commonly the first example mentioned when talking about the beauty of mathematics, and is almost ubiquitous with the idea of beauty in mathematics. It and its direct consequences are used regularly in calculus, engineering, physics.
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) + i\sin(\theta)\\ e^{i\theta}\cdot e^{-e\theta} &= (\cos(\theta) + i\sin(\theta))(\cos(\theta) - i\sin(\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 which lend themselves easily to taking complex arguments:
$ \begin{align} \cos(z) = \frac1{2}\left(e^{iz}+e^{-iz}\right)\\ \sin(z) = \frac1{2i}\left(e^{iz}-e^{-iz}\right) \end{align} $
Euler's equation has applications not only in mathematics, but also in engineering and physics. For example, signal processing used in Electrical Engineering. For signal processing, signals are best represented as a combination of sinusoidal functions) which can be easily expressed as the sum of complex exponential functions using euler's formula. As well, in physics, it is also used in phasor analysis to conceptualize the impedance of capacitors. In differential equations, we can see that euler's formula can be applied to simplify solutions because the complex exponential function is the eigenfunction.
Another Valuable result made simple by Euler's formula is De Moivre's Formula:
$ \begin{align} (\cos(\theta)+i\sin(\theta))^n &= \left(e^{i\theta}\right)^n\\ &= e^{i(n\theta)}\\ &= (\cos(n\theta)+i\sin(n\theta)) \end{align} $
"De Moivre's formula can be a convenient too for deducing multiple-angle trigonometric identities" (saff 29). It can also be a very useful tool in finding roots of unity (which have application to number theory (polynomials), geometry, etc.).
References
Cadwallader, C. (n.d.). Euler's Formula. Retrieved December 2, 2018, from https://www.projectrhea.org/rhea/index.php/HW1.3_Chris_Cadwallader_-_Eulers_forumla_ECE301Fall2008mboutin
Lee, D. (2011, December 8). On The Most Beautiful Equation. Retrieved December 2, 2018, from https://www.projectrhea.org/rhea/index.php/On_The_Most_Beautiful_Equation
Saff, E. B., & Snider, A. D. (2003). Fundamentals of Complex Analysis with Application to Engineering and Science (3rd ed.). Upper Saddle River, New Jersey: Pearson.
