sin2x2

sin2x2 is not a standard mathematical notation, so its meaning is ambiguous without explicit grouping or superscripts. In practice, the sequence could denote one of several common expressions, depending on how the terms are intended to be read.

One possible interpretation is sin(2x^2), the sine of twice the square of x. Here the argument is 2x^2, so the function is defined for all real x. Its value lies in the interval [-1, 1]. The function is even, since sin(2(-x)^2) = sin(2x^2). Its derivative, by the chain rule, is f'(x) = 4x cos(2x^2). The graph oscillates with a frequency that increases as |x| grows, due to the x^2 inside the sine.

Another common interpretation is sin^2(2x), the square of the sine of 2x. This interpretation yields values in [0, 1] and a period of π, since sin(2x) has period π. Using the identity sin^2(2x) = (1 − cos(4x))/2 can simplify some analyses. Its derivative is d/dx sin^2(2x) = 4 sin(2x) cos(2x) = 2 sin(4x). Graphically, it produces a wave with a nonnegative range and a smooth, repeating pattern.

If sin2x2 appears in a problem, the intended meaning should be confirmed by parentheses or superscripts. As conventions vary in plain text, writers often avoid this ambiguity by writing sin(2x^2) or sin^2(2x) explicitly. See also sine function, double-angle formulas, and trigonometric identities.