This Quantum World/Appendix/Sine and cosine

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Sine and cosine[edit]

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.