A catenary is the curve formed by a perfectly flexible chain suspended by its ends and acted on by gravity. Mathematically, it is described by the hyperbolic cosine function.
5 Must Know Facts For Your Next Test
The equation of a catenary in Cartesian coordinates is $y = a \cosh\left(\frac{x}{a}\right)$, where $a$ is a constant.
The shape of the catenary minimizes potential energy, making it an example of a variational problem.
Catenaries have applications in architecture and engineering, such as in the design of suspension bridges and arches.
In calculus, the arc length of a catenary can be determined using integration techniques involving hyperbolic functions.
The derivative of $\cosh(x)$ is $\sinh(x)$, which is useful in deriving properties related to the catenary.
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Related terms
Hyperbolic Cosine: $\cosh(x)$ is defined as $\frac{e^x + e^{-x}}{2}$ and describes the shape of a catenary.
Hyperbolic Sine: $\sinh(x)$ is defined as $\frac{e^x - e^{-x}}{2}$ and is the derivative of $\cosh(x)$.