Hyperbolic trigonometric functions — hyperbolic cosine, cosh(x) = (e^x + e^-x)/2, and hyperbolic sine, sinh(x) = (e^x - e^-x)/2 — are defined in terms of the exponential function and satisfy differentiation and identity relationships that closely parallel those of the circular (ordinary) trigonometric functions. This topic belongs to calculus, specifically to the study of special functions built from exponentials, and it connects to trigonometry through a structural analogy: just as cosine and sine parametrize the unit circle x^2 + y^2 = 1, cosh and sinh parametrize the right branch of the unit rectangular hyperbola x^2 - y^2 = 1, which is the origin of their names.
Hyperbolic trigonometric functions — hyperbolic cosine, cosh(x) = (e^x + e^-x)/2, and hyperbolic sine, sinh(x) = (e^x - e^-x)/2 — are defined in terms of the exponential function and satisfy differen…