cosh function
- Categories:
- hyperbolic functions
- pure mathematics
The cosh function is a hyperbolic function. It is also known as the hyperbolic cosine function.
Here is a video that explains sinh, cosh and tanh:
Equation and graph
The cosh function is defined as:
$$ \cosh{x} = \frac{e^{x}+e^{-x}}{2} $$
Here is a graph of the function:
cosh as average of two exponentials
The cosh function can be interpreted as the average of two functions, $e^{x}$ and $e^{-x}$. This animation illustrates this:
Other forms of the equation
If we multiply the top and bottom of the original equation for the cosh function by $e^{x}$ we get:
$$ \cosh{x} = \frac{e^{2x}+1}{2e^{x}} $$
Alternatively, if we multiply the top and bottom of the original equation for the cosh function by $e^{-x}$ we get:
$$ \cosh{x} = \frac{1+e^{-2x}}{2e^{-x}} $$
Derivation of other forms
To see how the two formulae above were derived, we start with the original definition of cosh:
$$ \cosh{x} = \frac{e^{x}+e^{-x}}{2} $$
Multiplying top and bottom by $e^{x}$ gives:
$$ \cosh{x} = \frac{e^{x}(e^{x}+e^{-x})}{2e^{x}} = \frac{e^{x}e^{x}+e^{x}e^{-x})}{2e^{x}} $$
Remember that $e^{x}e^{x}$ is $e^{2x}$. Also $e^{x}e^{-x}$ is 1. This gives:
$$ \sinh{x} = \frac{e^{2x}+1}{2e^{x}} $$
which is the second form of the cosh equation. The other alternative form is derived in a similar way.
See also
- sinh function
- tanh function
- sech function
- cosech function
- coth function
- arsinh function
- arcosh function
- artanh function
- Hyperbolic angle
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