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I was exploring hyperbolic functions and noticed something weird while comparing the analytical definition (e^x+e^-x)/2 with the geometrical definition using the hyperbola x^2 - y^2 = 1. For the angle pi/6, using the analytical definition gives approximately 1.14 while using the geometrical definition gives 1.22.

Does anyone know why?

Rohil Verma
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    how do you arrive at either of the numerical values? (telling us the software or calculator used, as well as what you asked it to compute, would help.) – user225318 Apr 11 '15 at 12:19
  • I used the windows calculator. To get 1.14 I just plugged pi/6 into the cosh function (verified by using the formula involving e) and I got 1.22 when I equated $x/root(3)$ to $root(x^2−1)$ – Rohil Verma Apr 11 '15 at 14:32
  • And why did you equate $x / \sqrt{3} = \sqrt{x^2 -1}$? Were you looking at some variant of this diagram? The value $a/2$ refers to the area of the shaded region, not the angle. – user225318 Apr 11 '15 at 17:19
  • Yes exactly that. I figured out the problem, the area and the angle are different. Is there some relation between them? – Rohil Verma Apr 11 '15 at 17:36
  • $$A(\theta) = \int_0^\theta (\cos^2 \theta - \sin^2\theta)^{-1/2}d\theta$$ Though if you evaluate that integral you will just get something involving the inverse hyperbolic functions. – user225318 Apr 11 '15 at 18:29

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