Vincos

Vincos is a mathematical function defined as the product of the sine and cosine of a real angle: vincos(x) = sin(x) · cos(x). Using the double-angle identity, this can also be written as vincos(x) = (1/2) sin(2x). The function is defined for all real numbers and has a range from -1/2 to 1/2.

Zeros occur when sin(x) = 0 or cos(x) = 0, i.e., at x = nπ/2 for integer n. Its derivative is vincos'(x) = cos^2(x) − sin^2(x) = cos(2x), so the critical points are at x = π/4 + kπ/2. At these points, vincos(x) equals (1/2)(-1)^k, yielding a maximum value of 1/2 and a minimum value of -1/2, alternating with period π.

The graph of vincos oscillates between -1/2 and 1/2, with maxima at x = π/4 + kπ and minima at x = 3π/4 + kπ. In a calculus or trigonometry setting, vincos serves as a convenient example to illustrate product-to-sum identities (sin x cos x = (1/2) sin 2x) and to practice simple integral calculations, such as ∫ vincos(x) dx = -1/4 cos(2x) + C. The function thus provides a compact, analytic instance of how sine and cosine interact in a multiplicative combination.