# Arc length

Trigonometry equalities, inequalities and expressions - sin, cos, tan, cot

### Arc length

Hello to you all. I need help with a math problem, and yehh its given me a headeach :/

The question

Show that the length s of the catenary between x=-b and x=b is decided by the expressions L=2 asinh(b/a), where sinhx is a the function sinus hyperbolicus x=e^x-e^(-x)/2

I know that the the arclength of \sinh(x) between a and b is given by http://www.mathwords.com/a/arc_length_of_a_curve.htm

But I have no idea what to do about x=e^x-e^(-x)/2 and L=2 asinh(b/a
Guest

### Re: Arc length

Use the fact that $$\cosh^2 x - \sinh^2 x = 1$$ (for most trigonometry identities there is a hyperbolic version usually with an additional minus sign wherever there is an explicit or implicit $$\sin^2 x$$ term). Also remember that $$\sinh x$$ differentiates to $$\cosh x$$ and $$\cosh x$$ differentiates to $$\sinh x$$.

The equation of a Catenary is $$y = a\ \cosh(x/a)$$. So
Arc length $$= \int_{-b}^b \sqrt{1+(dy/dx)^2}\quad dx$$
$$= \int_{-b}^b \sqrt{1+\sinh^2 (x/a)}\quad dx$$
$$= \int_{-b}^b \sqrt{\cosh^2 (x/a)}\quad dx$$
$$= \int_{-b}^b \cosh (x/a)\quad dx$$
$$= [a\ \sinh(x/a)]_{-b}^b$$
$$= a\ \sinh(b/a) - a\ \sinh(-b/a)$$
$$= 2a\ \sinh(b/a)$$
as desired.

Hope this helped,

R. Baber.
Guest

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