# This Quantum World/Appendix/Sine and cosine

#### Sine and cosine

We define the function $\cos(x)$ by requiring that

$\cos''(x)=-\cos(x),\quad \cos(0)=1$  and  $\cos'(0)=0.$

If you sketch the graph of this function using only this information, you will notice that wherever $\cos(x)$ is positive, its slope decreases as $x$ increases (that is, its graph curves downward), and wherever $\cos(x)$ is negative, its slope increases as $x$ increases (that is, its graph curves upward).

Differentiating the first defining equation repeatedly yields

$\cos^{(n+2)}(x)=-\cos^{(n)}(x)$

for all natural numbers $n.$ Using the remaining defining equations, we find that $\cos^{(k)}(0)$ equals 1 for k = 0,4,8,12…, –1 for k = 2,6,10,14…, and 0 for odd k. This leads to the following Taylor series:

$\cos(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!} = 1-{x^2\over2!}+ {x^4\over4!} -{x^6\over6!}+\dots.$

The function $\sin(x)$ is similarly defined by requiring that

$\sin''(x)=-\sin(x),\quad \sin(0)=0,\quad\hbox{and}\quad \sin'(0)=1.$

This leads to the Taylor series

$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!} = x-{x^3\over3!}+ {x^5\over5!} -{x^7\over7!}+\dots.$