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

Does anyone know the origin of the words $$\arccos$$, $$\arcsin$$ & $$\arctan$$? That is to say, why are they named like this? What connects "arc" with inverse?

• My guess would be: In the unit circle, arc length is the same as angle ($s = r\theta$ for $r=1$), so the "arc" would refer to the measure of the arc whose angle has that value as its cosine, for example.
– matt
Apr 15, 2011 at 17:58
• @matt: Exactly. In ancient (Hellenistic) times, angles were not given measures, everything was expressed in terms of lengths. The "standard" circle sometimes had oddball radius connected with $\pi$, in a foreshadowing of the idea of the radian. In late medieval Europe, the radius might be $10000$, or $1000000$, or an even larger power of $10$, since decimal fractions ("decimals") were not used. Apr 15, 2011 at 18:10

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 ...").

• On the other hand, one would rightly wonder why the arc- prefix is used for inverse hyperbolic functions... :) Apr 16, 2011 at 18:01
• @J.M. I've seen them more frequently with the arg- prefix, presumably as an apocope for "argument". Aug 21, 2011 at 18:54
• @Bruno: The notation I grew up with was $\mathrm{arsinh}$, $\mathrm{artanh}$... you get the idea. The $\arg$ stuff I saw much later, and the $\mathrm{arc}$ stuff even later than that... I still imagine someone meeting inverse hyperbolic functions for the first time (and presuming no knowledge of the, uhurm, complex relations between them and inverse circular functions) wondering why you'd see books using $\mathrm{arc}$. Aug 21, 2011 at 18:59
• In Italy, the inverse of sinh is settsinh. It means "the sector (settore in italian) whose hyperbolic sine is..." Aug 26, 2013 at 12:51

Sine comes from sinew- bowstring and is the measurement up a bow from a bowstring laid on a surface, to where an arrow (nocked at center) touches the bow. Cosine is the complementing measure from the archer's arm (w elbow placed at center point.) The circle is the unit of one forearm, the string length- one forward one back (+1 thru -1.) Since arc sine uses the bow vertically, I figured it was from using archer standing or the bow on a wall instead of a table. The archer's view, siting along arrow of measurement. Measures from the archer.

"Sine comes from sinew" I don't think so -- it comes from sinus, Latin for "bay", "pocket" or "curve." Compare to "sinuous"