Revisiting Pythagoras goes linear

Michael Lugo Uncategorized August 17, 2013 2 Minutes

Let’s say for some reason that you know $\sin \theta$ and $\cos \theta$ , for some angle $\theta$ , and you need to figure out what $\theta$ is. Let’s say, furthermore, that you live in some benighted age which doesn’t have calculators or even trigonometric tables.

There are a few approaches to this. One is what I did in my “Pythagoras goes linear” post. We can fit a linear model to the points

$(x_i, y_i) = (\cos \theta_i \sin \theta_i)$

where the $\theta_i$ range over an arithmetic sequence with endpoints 0 and $\latex \pi/2$, namely

$0, {\pi \over 2n}, {2\pi \over 2n}, \cdots, {n\pi \over 2n}.$

The model you get out is quite simple:

$\theta \approx \pi/4 - 0.7520 \sin \theta + 0.7520 \cos \theta$

This is actually just a few lines of R.

 n = 10^6
 theta = (c(0:n))*pi/(2*n);
 x = cos(theta); y = sin(theta);
 summary(lm(theta~x+y))

I’ll save you the output, but be impressed: r² = 0.9992$, and the mean square error is 0.0126 radians or 0.72 degrees. So we can write $\theta \approx \pi/4 + 0.7520 (-\cos \theta + \sin \theta)$ , in radians. In degrees this is $\theta \approx 45^o + 43.08^o(-\cos \theta + \sin \theta)$ .

For example, say you have an angle with $\sin \theta = 5/13, \cos \theta = 12/13$ — the smaller angle of a 5-12-13 right triangle. Then we get $\theta \approx 45^o + 43.08^o (-7/13) \approx 21.8^o$ . In fact $\theta = \tan^{-1} 5/12 \approx 22.6^o$ — we’re not off by much! And the error is never more than a couple degrees, as you can see in the plot below.

This was inspired by a post by Jordan Ellenberg that I came across recently: How to compute arctangent if you live in the 18th century, which refers back to my Pythagoras goes linear. A better approximation, although nonlinear, is

$\theta \approx (3y)/(2 + x)$

where $x = \sin \theta, y = \cos \theta$ . This is essentially a simplification of the rule that Ellenberg’s source (Hugh Worthington’s 1780 textbook, The Resolution of Triangles) gives, which can be translated into our notation as

$\theta = {86*pi \over 180} {y \over x/2 + 1}.$ Applying this to our test angle with $x = 12/13, y = 5/13$ , we get $\theta \approx 15/38 = 0.39474$ radians, while in truth $\tan^{-1} 5/12 = 0.39479$ .

This formula $\theta \approx (3y)/(2+x)$ is so nice that I can’t help but suspect there’s a simple derivation. Any takers?

Published by Michael Lugo

View all posts by Michael Lugo

Published August 17, 2013

4 thoughts on “Revisiting Pythagoras goes linear”

sjt says:

August 17, 2013 at 3:03 pm

I assume you mean $x = \cos\theta$ and $y = \sin\theta$ .

Here’s one way you could invent that approximation: you recall that
$\cos x \le \frac{\sin x}{x} \le 1$
and you’d like to know more precisely where in that range the middle function falls. Maybe you make the guess that it’s close to a convex combination of the two functions on either side:
$\frac{\sin x}{x} \approx (1-\lambda)\cos x + \lambda\cdot 1$
The choice $\lambda = \frac{2}{3}$ makes the $x^2$ terms equal, and then rearranging yields
$x \approx \frac{3\sin x}{2+\cos x}$
as desired.

(Sorry for any latex errors; there’s no preview, alas.)

Reply
logosintegralis says:

August 17, 2013 at 3:12 pm

This looks like it’s related to multivariable Padé approximants. Let me take quick stab at it, apologies for no LaTeX.

Assume that there is some approximating function of the form

t ~= (a0+a1*x+a2*y)/(1+b1*x+b2*y).

Multiply across and rewrite in terms of theta as:

t ~= a0 + (a1-b1*t) sin t + (a2-b2*t) cos t

Now, replace the trigonometric function with their series representations:

t ~= a0 + (a1-b1*t)[t – t^3/6 +O(t^5)] + (a2-b2*t)[1-t^2/2+t^4/24+O(t^6)]

Collecting terms up to t^4, we have

t ~= (a0+a2)t^0 + (a1 – b2) t + (-a2/2 – b1) t^2 +(-a1/6 + b2/2) t^3 + (a2/24 + b1/6)t^4

Now just match the powers and boom, five equations in five variables. Mathematica gives me a1 -> 3/2, b2 -> 1/2, and all others 0. Therefore, the final function form would be:

t ~= (3y/2)/(1+x/2) ~= 3y/(2+x)

and we’re done. Padé approximants are quite nifty and have cool applications. My personal favorite is using exp(x) ~= (2+x)/(2-x) to quickly calculate interest rates in my head.

logosintegralis

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logosintegralis says:

August 17, 2013 at 5:04 pm

Apologies, the x and y were switched in my last formula. As the first commenter pointed out, it should be x=cos t and y = sin t. I stand by the rest of the derivation, though.

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