Cosh
Calculate hyperbolic cosine (cosh) for any real number. Free online cosh calculator with exponential step-by-step results and interactive graph.
About This Calculator
The hyperbolic cosine function, denoted cosh(x), is a fundamental hyperbolic function defined as cosh(x) = (e^x + e^(-x)) / 2. It is the even counterpart to hyperbolic sine (sinh) and is an even function symmetric about the y-axis: cosh(-x) = cosh(x). Unlike regular cosine which oscillates between -1 and 1, cosh has a minimum value of 1 at x = 0 and grows exponentially as |x| increases.
Hyperbolic cosine is named from its relationship to a hyperbola in the same way that regular cosine relates to a circle. The parametric equations (cosh(t), sinh(t)) trace the right branch of the unit hyperbola x^2 - y^2 = 1. This geometric connection mirrors how (cos(t), sin(t)) traces the unit circle.
The cosh function appears in many real-world applications: it describes the catenary curve -- the natural shape of a hanging chain or cable supported at both ends (y = a·cosh(x/a)). This shape is used in suspension bridge design, power line installation, and architecture. The Gateway Arch in St. Louis follows an inverted catenary shape.
Frequently Asked Questions
What is hyperbolic cosine (cosh)?
Hyperbolic cosine (cosh) is a hyperbolic function defined as cosh(x) = (e^x + e^(-x)) / 2. Unlike regular cosine which is periodic, cosh is an even function with a minimum value of 1 at x = 0, growing exponentially as |x| increases. It describes the shape of a hanging cable (catenary).
How does cosh differ from regular cosine?
Regular cosine (cos) is periodic with period 2pi and ranges between -1 and 1. Hyperbolic cosine (cosh) is not periodic, has a minimum of 1 at x = 0, and grows without bound as |x| increases. While cos relates to the unit circle, cosh relates to a hyperbola.
What is the formula for cosh?
The exponential formula for hyperbolic cosine is: cosh(x) = (e^x + e^(-x)) / 2. Some common values: cosh(0) = 1, cosh(1) ≈ 1.5431, cosh(-1) ≈ 1.5431. It is an even function: cosh(-x) = cosh(x).
Where is hyperbolic cosine used in real life?
Hyperbolic cosine appears in physics (catenary curves for hanging cables and power lines), in engineering (suspension bridge design and arch supports), in architecture (the Gateway Arch in St. Louis follows a catenary shape), and in mathematics (solving differential equations).
What is the identity relating cosh and sinh?
The fundamental hyperbolic identity is cosh^2(x) - sinh^2(x) = 1, which is analogous to the Pythagorean identity cos^2(x) + sin^2(x) = 1 but with a minus sign. Other identities include cosh(2x) = cosh^2(x) + sinh^2(x) and d/dx cosh(x) = sinh(x).
Is this cosh calculator free?
Yes, all calculators on Calculy are completely free to use. No registration or download required.