# Confusion surrounding the geometric definitions of $\sinh$ and $\cosh$

$$\cos x$$ 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.

• Isn't this about $x$ and $y$ coordinates of points swept out on a hyperbola, with some suitable choice of parametrization? – paul garrett Sep 30 at 21:01
• 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. – brainjam Sep 30 at 21:40
• @brainjam It's also more natural to define radians in terms of area rather than angles. – CyclotomicField Sep 30 at 23:04
• 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. – Blue Oct 1 at 18:10