What is the deepest / most interesting known connection between Trigonometry and Statistics?

I'm teaching both at the same time to different classes in high school, so I just wondered about this.

Added by OP on 16.May.2011 (Beijing time)

1. I mean Statistics only, without Probability. In other words, Descriptive Statistics only. This rules out Buffon’s Needle Problem.

2. The occurrence of π is counted only as a connection to geometry. By Trigonometry is meant explicit, non-gratuitous, occurrence of the sine, cosine, tangent, or their reciprocals.

3. Yes, the Law of Cosines fits the bill, but it is on the surface: everyone knows about it. It would be a hugely interesting meta theorem that this were the “deepest” connection between Trigonometry and Descriptive Statistics. My suspicion/hope is that there are deeper connections, somewhat along the line of the surprising use of trig in solving the cubic in closed form, or the use of trigonometric substitutions in evaluating certain integrals. The comment below about the arcsine transformation at first blush seems to be something along this line, but when you follow the link you see that someone is bringing it up only to say how bad it is.

So, I hope the intent of my question is now much clearer.

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All I can think of is that the correlation coefficient can be expressed as a cosine, so that the correlation is maximal when cost is 1 , i.e., the two data vectors lie on the same line ( Given data sets {($x_1$,$y_1$),..,($x_n$,$y_n$)}, and {($x'_1$,$y'_1$),..,($x'_n$,$y'_n$)} (e.g, scores before- and after a prep test), then correlation is maximal when there is a (nonzero) constant k with k$y'_j$=$y_j$. Similar ideas are used in Factor Analysis and Principal Component Analysis. Is this what you had in mind? – gary May 15 '11 at 21:53
buffon needle is an example (or homework), nor deep or anything, just fun mste.illinois.edu/reese/buffon/buffon.html – yoyo May 15 '11 at 21:53
There is the so-called arcsine transformation... – J. M. May 16 '11 at 0:45

No doubt it's the Law of Cosines. The correlation between two data sets follows the generalized $n$-dimensional Law of Cosines.

EDIT: Maybe I'll make this a little more explicit. Take two data sets $A = (5,7,2...)$ and $B = (12, 4, 9...)$ and ask if they are correlated. One way is to treat them as vectors and look at the data set $C = A+B = (17, 11, 11...)$ where the sum of the data sets (vectors) is taken pointwise. Okay...it's not the length of the vectors that works like the Law of Cosines, but the standard deviation. If the two data sets are randomly correlated then you should expect the standard deviations to add like the Law of Pythagoras, so that if $\text{StDev}(A) = 3$ and $\text{StDev}(B)=4$ then $\text{StDev}(C)$ should equal $5$. For $100\%$ correlations, the standard deviation of $C$ would have to be $7$ (or $1$ for negative correlation). It's the Law of Cosines where the correlation is the cosine of the angle between two vectors of length $3$ and $4$.

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Here is a link to a similar answer to someone else's question, namely, asking for surprising generalizations: math.stackexchange.com/questions/1352/… – Mike Jones Jul 28 '11 at 5:22

The normal distribution has a $\pi$ in it. That's fairly deep.

Edit: I don't see how this doesn't count. $\pi$, after all, is $4 \arctan 1$, or half the period of the $\sin$ and $\cos$ functions. It is closely related to trigonometry and the properties of the trigonometric functions, and to call $\pi$ an occurrence of "geometry" as if that were something unrelated to trigonometry is mystifying.

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Michael Lugo's answer for the appearance of $e$ and $\pi$ in the normal distribution is is quite nice. – t.b. May 16 '11 at 11:32
Calling that occurrence of $\pi$ gratuitous is quite a stretch indeed! – Mariano Suárez-Alvarez May 16 '11 at 13:58

Perhaps this would be quite difficult to include in a high school class, but wouldn't Fourier analysis be a good example of this?

The characteristic function of a random variable which admits a density is just the Fourier transform of its density, and Fourier transforms are continuous versions of Fourier series which involve decomposition into sines and cosines. More explicitly, Fourier transforms involve exponentials of purely imaginary numbers which could also be written as trigonometric functions.

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You said no probability, but since another post mentions the normal distribution and you mentioned a desire for seeing a use as a trig substitution I think its worth considering Pearson distributions type IV which uses trig substitution that results in arctan. The Cauchy distribution is an example of a Type IV distribution which has applications to Chemistry and Physics. A version called the circular Cauchy distribution can be used to relate the random variable to an angle measure on the unit circle so there are all sorts of trig exercises that could be looked at using it.

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