Sinacosb

Sinacosb is a shorthand notation for the product sin(a) cos(b), where a and b are real numbers. The term appears in trigonometry and analysis when the two angles are treated as independent variables, such as in identities, integrals, or signal-processing formulas.

A central relation is the product-to-sum identity: sin a cos b = (1/2)[sin(a+b) + sin(a−b)]. This expresses sinacosb as a sum of sines. Derivatives with respect to the variables are ∂/∂a = cos a cos b and ∂/∂b = −sin a sin b. The expression is periodic in each variable with period 2π and has range [−1, 1]. It attains the maximum value 1 when a = π/2 + 2kπ and b = 2mπ, and the minimum value −1 when a = 3π/2 + 2kπ and b = 2mπ, for integers k, m.

Parities: sin a cos b is odd in a (sin(−a) cos b = − sin a cos b) and even in b (sin a cos(−b) = sin a cos b).

Applications: Sinacosb appears in Fourier analysis, trigonometric expansions, and signal-processing contexts involving modulated or mixed-angle terms. It also arises in the integration and simplification of products of sine and cosine functions.

See also: Sine function, Cosine function, Product-to-sum identities.