This Quantum World/Appendix/Sine and cosine

Sine and cosineEdit

We define the function   by requiring that

   and   

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

Differentiating the first defining equation repeatedly yields

 

for all natural numbers   Using the remaining defining equations, we find that   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:

 

The function   is similarly defined by requiring that

 

This leads to the Taylor series