sin4

sin4 is the evaluation of the sine function at the real number 4, using radians as the unit of angle. In mathematics, the sine function sin(x) takes real numbers to real numbers, with the standard convention that arguments are in radians unless specified otherwise. Therefore sin4 denotes sin(4 radians).

Numerical value: sin(4) ≈ -0.7568024953079282.

Properties and relations: The sine function is periodic with period 2π, so sin(4) = sin(4 − 2π) ≈ sin(-2.283185307179586). Using odd symmetry, sin(-θ) = -sin(θ), so sin(4) ≈ -sin(2.283185307179586). The value does not simplify to a known elementary constant.

Computation: sin(4) can be computed by the Taylor series sin x = x − x^3/3! + x^5/5! − x^7/7! + ..., substituting x = 4. The convergent series yields the same numerical value, and modern calculators and software use efficient algorithms to evaluate it.

Mathematical notes: For any nonzero algebraic number α, sin α is transcendental, so sin(4) is transcendental and cannot be expressed as a root of a polynomial with rational coefficients. In practice, it is treated as an irrational, non-elementary constant.

Applications: Values such as sin(4) appear in numerical experiments, signal processing, and the study of Fourier series, where sine values at various arguments are combined to approximate periodic functions.