Where is the 'c' in the spelling of the inverse hyperbolic function of sine? Isn't it just a hyperbolic version of $\operatorname{arcsin}(x)$? That is, why is it written $\operatorname{arsinh}(x)$ and not $\operatorname{arcsinh}(x)$?
The same question applies to the other functions as well.
As user90369 pointed out, the "arc" in "$\arcsin$" comes from Latin arcus, which is directly related to the English word "arc". This makes sense because functions like $\arcsin$ give you the length of the corresponding arc of the unit circle (which also happens to be twice the area of the corresponding sector).
In contrast, "ar" is short for "area", since the hyperbolic functions are related to areas bounded by the unit hyperbola. Specifically, $\mathrm{arsinh}$ gives you twice the area of the corresponding sector and isn't related to the arclength of the piece of the hyperbola in a simple way.
Wikipedia confirms this story with references to Mathematics: From the Birth of Numbers, Oxford Users' Guide to Mathematics, and Handbook of Mathematics.