sin2n

Sin2n, written as sin(2n), denotes the sine of the quantity 2n. In most mathematical contexts, the argument is measured in radians, and n may be any real or integer value. For real n, sin(2n) is a continuous function of n; for integer n, it samples the sine function at evenly spaced points with step 2.

Key identities and representations

- Double-angle form: sin(2n) = 2 sin n cos n.

- Complex/exponential form: sin(2n) = Im(e^{i2n}).

- Chebyshev form: sin(2n) = sin 2 · U_{n-1}(cos 2), where U_{n-1} is the Chebyshev polynomial of the second kind.

- Recurrence for integers: if s_n = sin(2n) and c_n = cos(2n), then s_{n+1} = s_n cos 2 + c_n sin 2 and c_{n+1} = c_n cos 2 − s_n sin 2, with s_0 = 0 and c_0 = 1.

Periodicity and distribution

- As a function of real n, sin(2n) has period 2π.

- When n is restricted to integers, the sequence n ↦ sin(2n) is not periodic because 2/π is irrational; the values do not repeat with a fixed period and, in a sense, are dense in the interval [−1, 1].

Computational notes

- Numerical evaluation uses standard sine function with radians. For many integer terms, the recurrence above provides an efficient way to generate successive values without repeated trig function calls.

See also

- Sine function, double-angle formula, De Moivre’s formula, Chebyshev polynomials.