$\cos x$ and $\sin x$ can be defined by rotating $x$ units around a unit circle: $\cos x$ is the $x-$coordinate and $\sin x$ the $y$-coordinate. However, I am struggling to understand the analogous definition for $\sinh x$ and $\cosh x$. I understand that a hyperbola can be defined as the set of all points satisfying $(\cosh t,\sinh t)$. However, this still begs the question as to how one can work out what $\cosh t$ and $\sinh t$ are in the first place. I know that the hyperbolic functions can be defined using exponentials, but I think a geometric interpretation would be nice.

  • $\begingroup$ Isn't this about $x$ and $y$ coordinates of points swept out on a hyperbola, with some suitable choice of parametrization? $\endgroup$ – paul garrett Sep 30 at 21:01
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    $\begingroup$ The Wikipedia article on hyperbolic functions explains the geometric interpretation. It's a little odd that the argument is an area, but in hyperbolic geometry area and angle are related. $\endgroup$ – brainjam Sep 30 at 21:40
  • $\begingroup$ @brainjam It's also more natural to define radians in terms of area rather than angles. $\endgroup$ – CyclotomicField Sep 30 at 23:04
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    $\begingroup$ Related (duplicate?): "Alternative definition of hyperbolic cosine without relying on exponential function". See, in particular, my answer. You may also be interested in the figures shown in this answer. $\endgroup$ – Blue Oct 1 at 18:10

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