tanRCx

TanRCx is a compact notation used in some mathematical and engineering contexts to denote the tangent of the product RCx, where R, C, and x are real quantities. The exact meaning depends on the setting, but a common interpretation is tanRCx = tan(R C x).

If R, C, and x are real numbers and RCx is dimensionless, then tanRCx = tan(R C x). The derivative with respect to x is d/dx tan(R C x) = R C sec^2(R C x). As a function of x, tanRCx has vertical asymptotes at x = (π/2 + kπ)/(RC) for integers k, provided RC ≠ 0. It is periodic with period π/(RC) and is an odd function, satisfying tanRC(-x) = -tanRCx.

In electrical engineering and physics, RC typically denotes the resistance-capacitance product, sometimes forming a time constant τ = RC. While tan(τ x) is not a standard circuit expression, tanRCx can appear in nonlinear or approximate models, or in analytical treatments involving phase angles and trigonometric substitutions. Its use is contingent on the context and the units involved, since RC must yield a dimensionless argument for the tangent function.

Example formulations: with R = 1 and C = 1, tanRCx reduces to tan(x). At x = 0.5, tanRCx ≈ tan(0.5) ≈ 0.5463. At x = π/(2RC) with RC = 1, tanRCx is undefined due to a vertical asymptote.

See also: tangent function, RC time constant, trigonometric functions.