cosh2

cosh2 commonly denotes the hyperbolic cosine evaluated at the real number 2, written cosh(2). By definition, cosh x = (e^x + e^{-x})/2, so cosh(2) = (e^2 + e^{-2})/2. This evaluates to approximately 3.7621956911.

Exact relations and identities help situate cosh(2) among hyperbolic functions. One key identity is cosh(2x) = cosh^2 x + sinh^2 x, which for x = 1 gives cosh(2) = cosh^2(1) + sinh^2(1). Another common form is cosh(2) = 2 cosh^2(1) − 1, derived from the double-angle formula. These identities reflect the close connection between cosh and sinh and their growth properties. The function cosh t is even and increases for t > 0; its derivative is sinh t, and its asymptotic growth behaves like (1/2) e^t for large t.

Applications of the hyperbolic cosine, including cosh(2), appear across mathematics and physics. In differential equations, hyperbolic functions arise as solutions to linear second-order equations with real coefficients. In physics, cosh(2) can appear in problems involving hyperbolic geometry, relativistic kinematics, or Laplace transforms. As a concrete constant, cosh(2) serves as a specific value in calculations and serves as a representative example of the broader properties of hyperbolic functions. Note that notation such as cosh^2(2) may denote (cosh(2))^2, so parentheses are important to avoid ambiguity.

---