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:

$\cos(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}=1-{x^{2} \over 2!}+{x^{4} \over 4!}-{x^{6} \over 6!}+\dots .$

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

$\sin(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}=x-{x^{3} \over 3!}+{x^{5} \over 5!}-{x^{7} \over 7!}+\dots .$