sin2kn

Sin2kn denotes the sine of the product 2kn, written as sin(2kn). In many contexts k is a constant real parameter and n is an integer index, so the expression describes a discrete-time sinusoid with angular frequency ω = 2k.

Interpretation and notation: When n is an integer, sin(2kn) forms a sequence in n that takes real values between −1 and 1. If k is expressed relative to π as k/π = p/q with integers p, q in lowest terms, the sequence sin(2kn) is periodic in n with period q. If k/π is irrational, the sequence is aperiodic.

Key identities: sin(2kn) can be written as 2 sin(kn) cos(kn). As a complex exponential, sin(2kn) = Im(e^{i2kn}). It also admits a representation in terms of exponentials: sin(2kn) = (e^{i2kn} − e^{−i2kn})/(2i).

Periodicity in k: For fixed integer n ≠ 0, the function of k given by sin(2kn) is π/n-periodic, since sin(2(k+π/n)n) = sin(2kn + 2π) = sin(2kn).

Applications: sin(2kn) appears in digital signal processing and Fourier analysis as a discrete-time sinusoid. It is used to model sampled sine waves, to study spectral content, and in solving recurrence relations and lattice sums where angular frequencies are scaled by n.

Examples: If k = π/6 and n = 1, sin(2kn) = sin(π/3) = √3/2. If k = π/2 and n = 3, sin(2kn) = sin(3π) = 0.