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.$$