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When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them immediately obvious. Most of the analogous identities involving $\sin,$ $\cos$ have such interpretations.

Is it possible to use hyperbolic geometry to prove identities in hyperbolic trigonometry geometrically? Any examples would be greatly appreciated (if this is in fact possible).

Addendum: If there is no analogy in hyperbolic trigonometry, could the complex plane and the relationship between hyperbolic and non-hyperbolic functions be used instead (i.e. $\cos(x)= \cosh(ix), \sin(x)=-i\sinh(ix))$?

The second identity is equivalent to saying that for all $x\in\mathbb{R}$ there is a triangle with hypotenuse $\cosh(x)$ and sides $\sinh(x),1$. Is there a geometric interpretation of $x$?

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I strictly dislike sending links to Wikipedia when it is about math and/or programming, but I remember that en.wikipedia.org/wiki/Hyperbolic_function has some nice illustrations on that topic. –  Evgeny Oct 14 '13 at 18:07
@Evgeny I agree that it is excellent, but it's all either algebra or Euclidean geometry (I could be an idiot in thinking that hyperbolic geometry has anything to do with hyperbolic functions, Euclidean hyperbolae could be the only geometrical interpretation). –  Alyosha Oct 14 '13 at 18:14
Illustrating (circular) trig identities with diagrams is aided by nature of similar triangles in Euclidean geometry. Drop a few perpendiculars here, add a parallel or two there, maybe inscribe an angle in a circle, and you can create lots of similar triangles to make your case. Non-Euclidean (in particular, hyperbolic) geometry lacks this utility. There are no similar-but-not-congruent triangles, so diagrams like ones you link just don't work. (The altitude to the hypotenuse of a right triangle does not create similar sub-triangles.) [continued] –  Blue Oct 14 '13 at 18:21
(Part 2) When you note that the Law of Sines in hyperbolic geometry is $$\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$$ you realize that the notion of proportionality of (raw) side-lengths is decidedly unhelpful; even the impact of addition and subtraction of lengths is unclear. Every geometric formula wraps lengths in some hyperbolic trig function ... which makes relations really hard (impossible?) to represent in intuitive visuals. In fact, this difficulty leads me to muse often about the question "What are trig classes like in a hyperbolic universe?" –  Blue Oct 14 '13 at 18:34
(Part 3) A final thought on this. Consider the Pythagorean Theorems in Euclidean and Hyperbolic geometries:$$a^2 + b^2 = c^2 \qquad \text{vs} \qquad \cosh a \; \cosh b = \cosh c$$ (I'm biased of course, but...) It seems that the notion of squaring numbers is fairly natural, so it's not too hard to accept that ancient geometers "noticed" the sum-of-squares formula. But what culture goes around cosh-ing numbers with such ease that the product-of-cosines is comparably "noticeable"? How does fluency in transcendental exponentials precede discovery of a (the?) fundamental geometric relation? –  Blue Oct 14 '13 at 18:55

1 Answer 1

Of course, they admit just the same interpretation in 1+1-dimensional pseudo-Euclidean space (with “dx2 − dv2” vector magnitude form) as trigonometric functions have in 2-dimensional Euclidean space. That pseudo-Euclidean space is also known as (the plane of) split-complex numbers and has its “trigonometric circle” with equation “x2 − v2 = 1” and hyperbolic angles. One minor difference: the “x > 0” inequality should be added to aforementioned equation to restrict trigonometric eh… hyperbola only to polar angles from −∞ to +∞.

Explanation with analytic functions on (standard) complex numbers is not mutually exclusive with metric signature argument presented above. If you consider ℂ2 = {(z, w)} where z = x + iy, w = u + iv, x, y, u, v ∈ ℝ, with quadratic form dz2 + dw2, then restriction to a totally real subspace of {(x, u)} with y = v = 0 gives Euclidean geometry with standard trigonometry, whereas restriction to the one of {(x, v)} with y = u = 0 gives aforementioned pseudo-Euclidean geometry with hyperbolic trigonometry. In other words, complexification of both 2-dimensional real spaces gives the same thing. Now you can write a parametric presentation of a line: $$ z = \frac{e^α + e^{-α}}2 t\\ w = i\frac{e^α - e^{-α}}2 t $$ and say that real α are hyperbolic angles (with corresponding lines lying in the pseudo-Euclidean plane for real t) and imaginary α are standard (circular) angles (likewise, in Euclidean plane). Note an identity $$(\frac{e^α + e^{-α}}2)^2 + (i\frac{e^α - e^{-α}}2)^2 = 1,$$ so t is (generally, complex) natural parameter: t2 = z2 + w2.

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