sinefunction

The sine function, denoted sin, is a fundamental trigonometric function. In a right triangle, for a given acute angle θ, sin θ equals the ratio of the length of the opposite side to the hypotenuse. In the unit circle interpretation, sin θ is the y-coordinate of the point on the circle of radius 1 obtained by rotating from (1,0) by angle θ.

Domain and range: The sine function is defined for all real numbers θ, with values in the interval [-1, 1].

Periodicity and symmetry: It is periodic with period 2π, and is an odd function: sin(-θ) = -sin θ.

Key values: sin 0 = 0, sin(π/2) = 1, sin π = 0, sin(3π/2) = -1.

Derivatives and integrals: The derivative of sin θ with respect to θ is cos θ; the integral is -cos θ + C.

Representations: The Taylor series expansion about 0 is sin θ = θ − θ^3/3! + θ^5/5! − ... . Euler's formula relates sin θ to exponentials: sin θ = (e^{iθ} − e^{−iθ})/(2i).

Applications: The sine function models periodic phenomena such as sound and light waves, alternating current, and other oscillations. It is essential in solving triangles, signal processing, Fourier analysis, and many areas of physics and engineering.