Etymology of $\arccos$, $\arcsin$ & $\arctan$?

Solution 1:

When measuring in radians, an angle of $\theta$ radians will correspond to an arc whose length is $r\theta$, where $r$ is the radius of the circle.

Thus, in the unit circle, "the arc whose cosine is $x$" is the same as "the angle whose cosine is $x$", because the measurement of the length of the arc of the circle is the same as the measurement of the angle in radians.

I'll note that in Mexico, the functions were also called $\mathrm{ang\,sin}$, $\mathrm{ang\,cos}$, etc., meaning "angle whose sine is..." and "angle whose cosine is..." (rather than "arc whose ...").