How did mathematicians discover the expressions of hyperbolic functions?

  • I
  • Thread starter Leo Liu
  • Start date
  • #1
112
47

Main Question or Discussion Point

The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are
$$
\begin{cases}
\sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x) \in \mathbb R \\
\cosh x = {e^x+e^{-x}} /2; \: x \in \mathbb R, \: f(x) \in [1,\infty)
\end{cases}
$$
But I would like to know how to derive these expressions. Thanks.

P.S. I do know the proof--##\cosh^2 t - \sinh^2 t = 1## is equivalent to ##x^2-y^2=1##.
 
Last edited:

Answers and Replies

  • #2
1,548
938
This would be easier to follow if you always call the parameter t and the Cartesian coordinates (x,y)
 
  • Like
Likes Leo Liu

Related Threads on How did mathematicians discover the expressions of hyperbolic functions?

Replies
5
Views
6K
  • Last Post
Replies
4
Views
4K
Replies
7
Views
587
  • Last Post
Replies
6
Views
20K
Replies
7
Views
1K
Replies
3
Views
610
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
1
Views
3K
Top