12sin2x

12sin2x is a mathematical expression commonly encountered in trigonometry and calculus. It represents the value of the sine function applied to twice the angle x, multiplied by the constant 12. The sine function, sin(θ), is a fundamental trigonometric function that relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse. The double angle identity for sine states that sin(2x) = 2sin(x)cos(x). Therefore, 12sin2x can also be written as 12 * (2sin(x)cos(x)), which simplifies to 24sin(x)cos(x). This expression appears in various contexts, including the analysis of wave phenomena, oscillations, and Fourier series. Its graph is a sinusoidal wave with an amplitude of 12 and a period of π. Understanding the behavior and manipulation of expressions like 12sin2x is crucial for solving trigonometric equations and integrating or differentiating trigonometric functions in calculus.