This Quantum World/Appendix/Sine and cosine

Sine and cosine 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