Sin2x

Sin(2x), written as sin(2x) or sin 2x, denotes the sine of the angle obtained by doubling x. It is a basic trigonometric function that arises in many double‑angle problems and identities. The standard double‑angle identity is sin(2x) = sin(x + x) = 2 sin x cos x.

In addition to the primary form, sin(2x) can be expressed as sin(2x) = 2 tan x / (1 + tan^2 x) for cos x ≠ 0. This follows from the tangent and Pythagorean identities. The function is an odd function, since sin(−2x) = −sin(2x).

Domain, range, and periodicity: the function is defined for all real x, with range [−1, 1]. It has period π, meaning sin(2(x + π)) = sin(2x). Zeros occur when 2x = nπ, i.e., x = nπ/2. Maximum values of 1 occur at x = π/4 + kπ, and minimum values of −1 occur at x = 3π/4 + kπ, for integers k.

Graph and interpretation: the graph of sin(2x) is a sine wave with twice the frequency of sin(x), completing a full cycle every half period of sin x. This makes it useful for modeling signals or quantities that oscillate with double the original frequency.

Applications: sin(2x) appears in solving trigonometric equations, integration and substitution with double angles, Fourier analysis, and signal processing, where it helps simplify expressions involving the sine of a sum or multiple of an angle.