Knowlpedia

Definition

Sine function

The odd trigonometric function defined by its entire power series.

The sine function is defined analytically by the everywhere convergent

sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!.\sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}.
Identities

It is odd, satisfies sin⁡0=0\sin0=0, and its derivative is the . The two series give sin⁡2x+cos⁡2x=1\sin^2x+\cos^2x=1 and the addition formulas.

Angle convention

The constant is the least positive zero of sine and fixes its radian angle convention. With this scale, (cos⁡θ,sin⁡θ)(\cos\theta,\sin\theta) parametrizes the unit circle and has period 2π2\pi.

Derivatives

Termwise differentiation yields sin⁡′′x=−sin⁡x\sin''x=-\sin x, so the function is smooth and solves this second-order equation with initial values sin⁡0=0\sin0=0, sin⁡′0=1\sin'0=1.

References