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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.

  • What makes a number special? Is $\frac 1 2\left(e^{\pi}+e^{-\pi}\right)$ special? Is a special number an algebraic number? – Git Gud Jun 28 '14 at 17:50
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    Yes, $\cosh(42)=\dfrac{\mathrm{e}^{42}+\mathrm{e}^{-42}}2$. – gniourf_gniourf Jun 28 '14 at 17:51
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    After the edit I still honestly don't understand what kind of numbers you're looking for. It seems to me that you're after algebraic numbers, but if that were the case you'd say so. How is, for example $e^{42}$, not a real expression written in terms of basic operations? – Git Gud Jun 28 '14 at 18:31
  • @GitGud The "paper" cited as a good example deals with hyperbolic functions of $r_1\ln r_2$ with $r_1,r_2\in\mathbb{Q}$ :) – Start wearing purple Jun 28 '14 at 18:34
  • @GitGud Sorry for confusion. I mean any number formed by operations on integers of addition, subtraction, multiplication and division, and extraction of $n$-th roots. –  Jun 28 '14 at 18:37
  • @O.L. I want the output to be simple, I am not as concerned about the input. Notice that in $\sin \frac{\pi}{2}$ the input is transcendental. –  Jun 28 '14 at 18:38
  • @GitGud Basically. There are some (like roots of high-order polynomials) which can't be written that way (e.g. "root of $x^{17}-14x^2+5$") but that is the idea. –  Jun 28 '14 at 18:40
  • @NotNotLogical Actually I take back what I said. What you describe are algebraic numbers. Of course there are algebraic numbers which can't be obtained that way, but the ones you described are algebraic. – Git Gud Jun 28 '14 at 19:13
  • replacing $\phi$ (in the linked note) with other real quadratic units might produce similar results. – yoyo Jan 19 '17 at 02:24

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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?

M. Winter
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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.

yoyo
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  • Why bother with the $d$? I mean I get the relation to the Pell equation, but the content of your answer appears to be more to do with the fact that $x^{2}-y^{2}=1$ is the equation of a hyperbola in the plane, and in particular $\cosh$ and $\sinh$ parametrize (a branch of) this curve (a bit like how $\cos$ and $\sin$ parametrize the unit circle $x^{2}+y^{2}=1$). – Will R Jan 19 '17 at 23:39