In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use the natural exponential function e x. And are not the same as sin(x) and cos(x), but a little bit similar: sinh vs sin. cosh vs cos. Catenary

The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle \((x = \cos t\) and \(y = \sin t)\) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations:

Hyperbolic Trigonometry Trigonometry is the study of the relationships among sides and angles of a triangle. In Euclidean geometry we use similar triangles to define the trigonometric functions—but the theory of similar triangles in not valid in …

Our primary objective is to discuss trigonometry in the context of hyperbolic geometry. So we will need to consider the hyperbolic trigonometric functions and how they relate to the traditional circular trigonometric functions. In particular, we will introduce the angle of parallelism in hyperbolic geometry, which provides

Hyperbolic functions are defined analogously to trigonometric functions. We have main six hyperbolic functions, namely sinh x, cosh x, tanh x, coth x, sech x, and cosech x. They can be expressed as a combination of the exponential function.

2024年11月25日 · Learn the different hyperbolic trigonometric functions, including sine, cosine, and tangent, with their formulas, examples, and diagrams. Also, learn their identities.

2025年1月20日 · The hyperbolic functions sinhz, coshz, tanhz, cschz, sechz, cothz (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing is appearing in the complex exponentials.

2023年8月29日 · All six hyperbolic functions can now be defined in general, analogous to the trigonometric (circular) functions: The graphs of the hyperbolic functions are shown below:

Unlike trig functions, hyperbolic functions are not periodic! These identities do not require Pythagoras' theorem, they can be derived from the de nition with a direct calculation and using properties of the exponential.