sin0

sin0 is shorthand for the sine of zero, written as sin(0) in mathematical notation. The expression evaluates to 0. In the standard interpretation, the sine function sin(x) maps real numbers x (measured in radians) to the y-coordinate of the point on the unit circle at angle x from the positive x-axis. At x = 0, the point is (1,0), so sin(0) = 0.

Properties: The sine function is an odd function, so sin(-x) = -sin(x). It is differentiable everywhere, with derivative cos(x); in particular, the derivative at 0 is cos(0) = 1. The Maclaurin series for sin x is x − x^3/3! + x^5/5! − ...; consequently, lim_{x->0} sin x / x = 1.

Zeros: sin(x) = 0 at x = k*pi for integers k. Thus sin(0) = 0 and sin(n*pi) = 0 for all integers n. Graphically, the sine wave crosses the origin and has zeros at multiples of pi.

Applications: The value sin(0) = 0 appears in many contexts, including trigonometric identities, Fourier analysis, and solutions to simple harmonic motion. While sin(0) itself is a single value, the surrounding properties of sin(x) near zero are used for approximations and series expansions.

Notes: In many texts, sin0 and sin(0) are used interchangeably; the angle is almost always treated in radians for calculus and analysis. The numerical value is exactly zero.