# Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals?

(My personal "feel" is that $\sin$ and $\cos$ are first-class citizens, $\tan$ is "1.5th-class," and the rest are second-class; I'm sure there are others who feel the same.)

Main question(s): From a purely high-school-geometric/"ninth-century-geometer's" standpoint, is there any reason why this should be so? Given the usual elementary knowledge of triangles when one is first introduced to these functions, I think it appears pretty arbitrary. How should I convince a high school student that $\sin$, $\cos$, and $\tan$, instead of their reciprocals, should be our main objects of study? How did history decide on their superiority?

Of course, with real analysis goggles, things look quite a bit different: $\sin$ and $\cos$ are the only ones that are continuous everywhere; $f'' = -f$ characterizes all their linear combinations; they have much nicer series representations; etc. But I suspect this is all hindsight.

(I don't pretend to know enough about complex analysis, but I suspect even more nice things happen there with $\sin$ and $\cos$, and even more ugly things happen with the other four. In any case, I doubt history chose $\sin$ and $\cos$ to be first-class citizens because of their complex properties.)

Secondary questions: Is there any reason why $\sin$ is the "main" function and $\cos$ is "only" its complement, or is this arbitrary as well? Is there any reason why $\tan$ is preferred to $\cot$?

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Circles. ${}{}$ –  anon Jul 25 '12 at 6:34
In particular, coordinates of points on circles. –  Thomas Andrews Jul 25 '12 at 6:36
$\tan \theta$ is the slope of a line at angle $\theta$ from horizontal; we prefer $\tan$ to $\cot$ because we prefer $\dfrac{\text{rise}}{\text{run}}$ to $\dfrac{\text{run}}{\text{rise}}$, perhaps. –  Jonas Meyer Jul 25 '12 at 6:41
(I've always hated that the inversion of $\cos$ is $\sec$ and the inversion of $\sin$ is the "cosec". Bugs me that there is this inversion of which is primary and which is the "co-". It feels arbitrary.) –  Thomas Andrews Jul 25 '12 at 6:51
@ThomasAndrews Well, one nice thing about how the co-functions are defined is when it comes to derivatives. The derivatives of the co-functions all sport a minus sign. –  Mike Jul 25 '12 at 7:01

Historically, trigonometric functions were originally computed in terms of "chords" (the angles subtended by a circular arc), and the sine was computed from a bisected chord (so a "half-chord"). In fact, the word "sine" originates from a mis-translation (from Arabic) of the Sanskrit word "jyā" which literally means "bow-string".

(The word "jyā" became "jība" in Arabic, and then subsequently "jaib". A cognate of "jaib" in Arabic has the meaning of "bosom", and jaib was mistakenly rendered into Latin as "sinus". So yes, the word "sine" was originally not safe for work).

The importance of the cosine seems also to have been first recognized by Indian mathematicans, who called it "koṭi-jyā" or "kojyā" meaning (roughly) "sine of the extreme angle" ("koṭi" means "the extreme end of a bow" or "extremity" in general).

So the sine as the "main" function, and cosine as the "adjunct" function goes back at least 16 centuries (in the Surya Siddhanta, written some time in the 4th century).

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My goodness, Mr Wheeler, how do you know this stuff? Very impressed! –  user22805 Jul 25 '12 at 8:31
Fun fact: In some languages, in particular Spanish, sine still has the same not safe for work meaning. This has lead to uncountable repressed giggles in high school math class. –  Javier Jul 25 '12 at 14:48

To elaborate on anon's comment, the classic trigonometric diagram has a right triangle drawn inside a unit circle, with one point at the origin, one point on the circle, and the third point the projection of the second onto the $x$ axis. Sine and cosine then give the lengths of the legs. I suppose it's more "natural" to draw the hypotenuse as having fixed unit length and studying how the legs change as you rotate the point around a circle, than to fix one of the legs.

Of course, which functions are "first-class citizens" really depends on the application; in my area (computer graphics, and in particular physical simulation) the first-class function is $2\tan \frac{\theta}{2}$, since it is easily computed from dot and cross products, bijective between $(-\pi,\pi)$ and the reals, and is the identity to first order near $\theta=0$.

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Even better, $2\tan\frac{\theta}{2}$ is the identity to second order near $\theta=0$ (like $\sin$ and $\tan$). –  Jonas Meyer Jul 25 '12 at 7:11
Funny, I can't recall encountering $2\tan\frac\theta2$ very much at all in my graphics/simulation code. Can you name some examples? –  Rahul Jul 25 '12 at 8:07
Here are a few: bending energy of rods and plates, the constraint function for enforcing that two angles are equal, mean value coordinates, some formulations of mean curvature, modeling circular arcs with NURBS (via Weierstrass substitution). On reflection $\cot$ might give $\tan \frac{\theta}{2}$ some competition: the ubiquitous "cotan Laplacian," Wachspress coordinates, etc. –  user7530 Jul 25 '12 at 23:46