This Quantum World/Appendix/Sine and cosine

Sine and cosine

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

${\cos }^{″}(x)=-\cos (x),\cos (0)=1$  and  ${\cos }^{\prime }(0)=0.$

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:

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

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

${\sin }^{″}(x)=-\sin (x),\sin (0)=0,\text{and}{\sin }^{\prime }(0)=1.$

This leads to the Taylor series

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