$dy/dx=\mathrm{sinh}(x)$ A tangent line through the origin has equation $y=mx$. If it meets the graph at $x=a$, then $ma=\mathrm{cosh}(a)$ and $m=\mathrm{sinh}(a)$. Therefore, $a\mathrm{sinh}(a)=\mathrm{cosh}(a)$.

Use Newton's Method to solve for $a$

Use Newton's Method to solve for $a$