My friend told me that $\sinh$ and $\cosh$ result from an exponential function, but I can't figure out why
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[Wikipedia](https://en.wikipedia.org/wiki/Hyperbolic_functions#Definitions) should answer your question. – DMcMor Jan 06 '21 at 16:10

See https://en.wikipedia.org/wiki/Hyperbolic_functions#Exponential_definitions – Gary Jan 06 '21 at 16:10

There's nothing to "figure out", just look at the definition. – Jan 06 '21 at 16:14

The tricky aspect to this question is what definition you're using of the hyperbolic trig functions. In fact, the most obvious definition is in terms of $e^x$ (as Kenny's answer below indicates). By contrast, it's slightly more interesting if you define the hyperbolic functions in terms of differential equations. – Semiclassical Jan 06 '21 at 16:15
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$$\cosh(x) = \frac{1}{2} (\exp(x)+\exp(x))$$ $$\sinh(x) = \frac{1}{2} (\exp(x)\exp(x))$$
$\exp(x)$ is the natural exponential function $\exp(x)=e^x$
Kenny
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