sine

The sine function, denoted sin(θ), is a trigonometric function that assigns to an angle θ the ratio of the length of the side opposite θ to the hypotenuse in a right triangle. On the unit circle, sin(θ) is the y-coordinate of the point obtained by rotating (1,0) by θ radians around the origin.

Sine is defined for all real numbers and takes values between -1 and 1. It is periodic with period 2π and is an odd function, meaning sin(-θ) = -sin(θ). It is closely related to the cosine function by the identity sin(θ) = cos(π/2 − θ). Its derivative is cos(θ), and its integral is −cos(θ) + C.

The function can be represented in several ways. Its Taylor series expansion is sin(θ) = θ − θ^3/3! + θ^5/5! − ..., valid for all real θ. In complex form, Euler's formula e^{iθ} = cos(θ) + i sin(θ) leads to sin(θ) = (e^{iθ} − e^{−iθ})/(2i).

Sine plays a central role in modeling periodic phenomena such as sound and light waves, alternating current, and mechanical vibrations. It is foundational in Fourier analysis, signal processing, and the solution of differential equations in physics and engineering.

History and etymology: The concept originated in Indian astronomy as the half-chord function jyā-ardha. It was transmitted to the Islamic world and later to Europe, where it was latinized as sinus. The English word sine comes from Latin sinus, reflecting an intermediate Arabic term jaīb for the half-chord, itself linked to jyā-ardha.