The circular trig function can be used to solve triangles in the Euclidean plane. Can the hyperbolic trig functions be used in any similar way?
2026-03-29 04:11:45.1774757505
Hyperbolic versus circular trigonometry
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Not strictly mathematics, but hyperbolic functions enter into special relativity.
Suppose that an observer sees two objects moving away from him in the same direction, object 1 at speed $v_1$ and object 2 at a greater speed $v_2$. If the speed of light is called $c$, then how fast (meaning relative speed $\Delta v$) does someone riding on object 1 see object 2 moving ahead of him?
In pre-relativistic mechanics, we would have simply
$\Delta v=v_2-v_1$.
But with the proper relativistic theory (in the absence of gravity, which is its own story) the relationship involves the hyperbolic tangent function:
$\Delta v/c=\tanh[\tanh^{-1}(v_2/c)-\tanh^{-1}(v_1/c)]$.
The hyperbolic tangent relation comes from the (Lorentz) transformation between spacetime reference frames being a modified form of a rotation.