1.54308063
2.71828183
0.36787944
1.54308063
1.54308063
2.71828183
0.36787944
1.54308063
The Hyperbolic Cosine Calculator computes $$\cosh(x)$$ for any real number $$x$$. The hyperbolic cosine is one of the fundamental hyperbolic functions, closely related to exponential functions and appearing throughout physics, engineering, and architecture.
The hyperbolic cosine is defined as:
$$\cosh(x) = \frac{e^x + e^{-x}}{2}$$
Notice the key difference from $$\sinh(x)$$: where hyperbolic sine uses subtraction, hyperbolic cosine uses addition. This sum of exponentials produces a function that is always positive and has its minimum value of 1 at $$x = 0$$.
Perhaps the most famous application of $$\cosh$$ is the catenary — the curve formed by a uniform chain or cable hanging freely under gravity. The equation of a catenary is:
$$y = a\cosh\left(\frac{x}{a}\right)$$
where $$a$$ is a parameter related to the tension and weight per unit length. The Gateway Arch in St. Louis is a weighted catenary, and many suspension bridge cables approximate catenary shapes. Galileo originally guessed that the hanging chain formed a parabola, but Leibniz, Huygens, and Johann Bernoulli proved in 1691 that it is actually a hyperbolic cosine.
The hyperbolic cosine satisfies the identity:
$$\cosh^2(x) - \sinh^2(x) = 1$$
This is the hyperbolic Pythagorean identity, directly analogous to $$\cos^2(x) + \sin^2(x) = 1$$. Note the minus sign instead of plus, reflecting the hyperbolic (rather than circular) geometry.
Beyond the catenary, $$\cosh$$ appears in solutions to the wave equation, heat equation, and Laplace's equation in rectangular coordinates. In electrical engineering, the characteristic impedance and propagation along transmission lines involve $$\cosh$$. In structural engineering, the deflection of a uniformly loaded cable follows a cosh profile. The function also appears in the Lorentz factor expression in special relativity when written in terms of rapidity: $$\gamma = \cosh(\phi)$$. In differential geometry, $$\cosh$$ describes the metric of hyperbolic space, which has applications in modern physics and even in machine learning (hyperbolic embeddings).
Enter any real number $$x$$. The calculator evaluates $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$ and displays the intermediate exponential values $$e^x$$ and $$e^{-x}$$ for transparency.
The result $$\cosh(x)$$ is always at least 1, with the minimum occurring at $$x = 0$$. The function grows exponentially for large $$|x|$$. Since $$\cosh$$ is an even function, $$\cosh(x) = \cosh(-x)$$, so the output is the same for positive and negative inputs of equal magnitude.
Inputs
Results
cosh(0) = (e⁰ + e⁰)/2 = (1 + 1)/2 = 1. This is the minimum value of cosh.
Inputs
Results
cosh(2) = (e² + e⁻²)/2 = (7.389 + 0.135)/2 ≈ 3.762.
Since $$e^x > 0$$ and $$e^{-x} > 0$$ for all real $$x$$, their sum is always positive. By the AM-GM inequality, $$\frac{e^x + e^{-x}}{2} \ge \sqrt{e^x \cdot e^{-x}} = \sqrt{1} = 1$$. Equality holds only when $$e^x = e^{-x}$$, i.e., at $$x = 0$$.
A catenary is the curve formed by a uniform chain hanging under gravity. Its equation is $$y = a\cosh(x/a)$$, where $$a$$ depends on the chain's tension and weight. Famous structures like the Gateway Arch and power line cables follow catenary curves.
For large $$|x|$$, $$\cosh(x) \approx \frac{e^{|x|}}{2}$$ because one exponential term dominates. In fact, $$\cosh(x) = \frac{e^x + e^{-x}}{2}$$ is exactly the average of $$e^x$$ and $$e^{-x}$$.
The inverse hyperbolic cosine is $$\text{arccosh}(x) = \ln\left(x + \sqrt{x^2 - 1}\right)$$ for $$x \ge 1$$. The domain restriction $$x \ge 1$$ reflects the fact that $$\cosh$$ never drops below 1.
No. $$\cos(x)$$ oscillates between -1 and 1, while $$\cosh(x)$$ grows exponentially and is always $$\ge 1$$. They are related by the identity $$\cosh(x) = \cos(ix)$$, where $$i$$ is the imaginary unit.
$$\cosh(a + b) = \cosh(a)\cosh(b) + \sinh(a)\sinh(b)$$ and $$\cosh(2x) = 2\cosh^2(x) - 1 = 1 + 2\sinh^2(x)$$. These mirror the standard cosine addition formulas with sign adjustments.
Roboculator Team
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