2cosx

2cosx denotes the function f(x) = 2 cos x, where cos x is the standard cosine function. The domain is all real numbers and the range is [-2, 2], since cos x itself lies in [-1, 1].

As a trigonometric function, 2 cos x is periodic with period 2π and is even, meaning f(-x) = f(x). It attains its maximum value 2 when cos x = 1 (e.g., x = 2kπ) and its minimum value -2 when cos x = -1 (e.g., x = π + 2kπ). The zeros occur where cos x = 0, at x = π/2 + kπ for integers k.

A common alternative expression uses Euler’s formula: cos x = (e^{ix} + e^{-ix})/2, so 2 cos x = e^{ix} + e^{-ix}. This representation links the function to complex analysis and is convenient in solving trigonometric equations, Fourier analysis, and when converting between exponential and trigonometric forms. The function also appears naturally in the study of rotation matrices, where the trace of a 2D rotation matrix is 2 cos θ.

Related ideas include identities involving multiple angles and Chebyshev polynomials, since cos(nθ) can be written as a polynomial in cos θ. In many contexts, 2 cos x serves as a basic building block for expressing sums of complex conjugates and for modeling oscillatory phenomena in physics, engineering, and signal processing.