sinh2

Sinh2 can be read in two related ways in hyperbolic trigonometry. First, sinh(2) denotes the hyperbolic sine evaluated at the argument 2. It has the exact form (e^2 − e^−2)/2 and, as a constant, approximately equals 3.6268604078.

Second, sinh^2 x denotes the square of the hyperbolic sine function. It is defined by sinh^2 x = [ (e^x − e^−x)/2 ]^2 = (e^{2x} − 2 + e^{−2x})/4. The function sinh^2 x is even, with domain all real numbers and range [0, ∞). It relates to other hyperbolic functions through identities such as sinh^2 x = (cosh 2x − 1)/2 and cosh 2x = 1 + 2 sinh^2 x. A further connection is tanh^2 x = sinh^2 x / cosh^2 x.

These relationships arise from the exponential definitions of the hyperbolic functions and mirror analogous trigonometric identities. The square of the hyperbolic sine is nonnegative for all real x and grows roughly as e^{2x}/4 for large positive x and as e^{−2x}/4 for large negative x.

Applications of sinh^2 x appear in calculus, especially in integrals and differential equations involving hyperbolic functions, as well as in physics contexts that use hyperbolic geometry or relativistic formulations. The fixed value sinh(2) is used when a specific numeric input is required, while sinh^2 x serves as a building block in expressions and transformations that involve hyperbolic trigonometric identities.