The constant pi - Knowlpedia

The constant $\pi$ is the least positive zero of the sine function defined by its power series. Thus $\sin\pi=0$ and $\sin x>0$ for $0<x<\pi$ . This definition fixes the usual radian scale without first assuming a geometric angle convention.

Existence and period

Cosine has a first positive zero $c$ . To see existence, if cosine stayed positive, sine would increase and be bounded below by a positive constant after some positive time. Then $\cos'=-\sin$ would force cosine to become negative, a contradiction. Continuity gives a least positive zero, with cosine positive before it.

The identities $\sin^2c+\cos^2c=1$ and $\sin'>0$ before $c$ give $\sin c=1$ . Addition formulas show $\sin(2c)=0$ , and $\sin(c+y)=\cos y>0$ for $0<y<c$ . Hence $\pi=2c$ . Also $\cos\pi=-1$ , and sine and cosine have period $2\pi$ . Their unit-circle parametrization has unit speed and one full turn over that period, giving the usual circumference-to-diameter interpretation.