There are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the hyperbolic trig functions?
The output should be expressible as sums, differences, products, quotients and $n$-th roots of integers. This paper gives some examples of what I am talking about.
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$\begingroup$As $\sin$/$\cos$ work well with $\pi$, so do $\sinh$/$\cosh$ work well with $e\approx 2.71828$, or more precisely, with the logarithm $\ln(\cdot)$ to the base $e$. E.g.
$$\cosh(\ln(x))=\frac{x+1/x}2=\frac{x^2+1}{2x}.$$
Is $\cosh(\ln(2))=5/4$ a special value?
$\endgroup$ $\begingroup$If $x^2-dy^2=1$ say, then $$ \sinh(\log(x+\sqrt{d}y))=\frac{1}{2}\left(x+\sqrt{d}y-\frac{1}{x+\sqrt{d}y}\right)=\frac{1}{2}\left(x+\sqrt{d}y-(x-\sqrt{d}y)\right)=\sqrt{d}y, $$ $$ \cosh(\log(x+\sqrt{d}y))=\frac{1}{2}\left(x+\sqrt{d}y+\frac{1}{x+\sqrt{d}y}\right)=\frac{1}{2}\left(x+\sqrt{d}y+x-\sqrt{d}y\right)=x, $$ and similarly with other integral Pellian equations.
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