# Trig reciprocal function nomenclature?

The fact that the reciprocal of $\sin\theta$ is $\csc\theta$, and the reciprocal of $\cos\theta$ is $\sec\theta$ messed with my head for the longest time when I was taking trig. Why are the functions named this way, when an alliterative scheme would seemingly be more sensible?

While I'm at it, what's the reason for choosing the names sine, cosine, and tangent? The words sinusoidal and tangental come to mind, but perhaps these words could have come from the function names, not the other way around.

-
The properties of secant align better with the properties of sine than the properties of the cosine. For example, if you look at the formulas for the derivatives, the derivatives of sine, tangent, and secant have no minus sign, while the derivatives of cosine, cotangent, and cosecant all have a minus sign in them. – Arturo Magidin Nov 28 '11 at 3:27
For the etymology, you can see Wikipedia. "Sinusoidal" is derived from "sine"; sine is derived from the Sanskrit word for "chord" (via Arabic), since sines were originally related to lengths of certain chords in circles. Tangent comes from the Latin for "touching", and it is also related to circles (and tangent lines to the circles). – Arturo Magidin Nov 28 '11 at 3:30
Throw in versine, vercosine, haversine, coversine, hacovercosine, and excosecant, and you get a party... – Srivatsan Nov 28 '11 at 3:55

Here is a nice explanation of the origin of all these names. In summary:

Most of the words come from Latin descriptions of the geometry involved. Sine comes from the Latin word 'sinus', tangent from the Latin 'tangens', and secant from the Latin 'secans'. The origin of the co-functions actually makes quite a lot of sense; cosine was originally co-sine, referring to the sine of the complementary angle. Similarly, cotangent and cosecant are the tangent and secant of the complementary angle.

I recommend reading the webpage if you want to know more detail, it's quite interesting.

-

These two pictures, which I created using xfig in 2008 for this Wikipedia article, may explain the nomenclature.

Vertical lines don't have "co-"; horizontal lines do.

-
Great tool for visualization! Thanks! – TreyK Nov 29 '11 at 4:05