sinx2x

The trigonometric identity sin(x)2x, more formally written as sin(2x), represents a fundamental relationship in trigonometry. This identity states that the sine of an angle that is twice another angle, x, is equal to twice the product of the sine of that angle and the cosine of that angle. Mathematically, it is expressed as sin(2x) = 2sin(x)cos(x).

This double angle formula is derived from the angle addition formula for sine, which is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). By setting a = x and b = x, we arrive at sin(x + x) = sin(x)cos(x) + cos(x)sin(x), which simplifies to sin(2x) = 2sin(x)cos(x).

The sin(2x) identity has numerous applications in calculus, physics, and engineering. It is frequently used to simplify trigonometric expressions, solve trigonometric equations, and in the evaluation of integrals. For instance, in calculus, it can be employed to transform integrands into simpler forms that are easier to integrate. In physics, it might appear in the analysis of wave phenomena or oscillatory motion. Understanding and applying this identity is crucial for anyone working with trigonometric functions.