sinxy

The term sinxy is not a single, universally defined mathematical object. In most contexts it is shorthand used informally to denote one of two common forms involving the sine function: either the sine of the product of two variables, sin(xy), or the product of sines, sin x · sin y. Because the notation is not standardized, its meaning should be inferred from context or explicitly defined in a given text or code.

If sinxy is interpreted as sin(xy), then it is the composition of the sine function with the product of two variables. This is a smooth, analytic function of two real variables. Its partial derivatives are straightforward: ∂/∂x sin(xy) = y cos(xy) and ∂/∂y sin(xy) = x cos(xy). Its Taylor series around a point, for example near (0,0), begins as sin(xy) = xy − (xy)^3/3! + (xy)^5/5! − … . There are no simple product-to-sum identities for sin(xy) involving only sin x and sin y, since the argument is the product, not a sum of angles.

If sinxy is read as sin x sin y, then it is the product of the sines of two angles. A standard identity applies: sin x sin y = [cos(x − y) − cos(x + y)]/2. This form is useful in Fourier analysis and trigonometric expansions. Its partial derivatives are ∂/∂x = cos x sin y and ∂/∂y = sin x cos y. The function is symmetric in x and y and periodically dependent on each variable with period 2π.

Ultimately, the intended meaning of sinxy should be specified by the author or software in use, since the symbol alone is ambiguous.

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