Let’s say for some reason that you know and , for some angle , and you need to figure out what 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

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

The model you get out is quite simple:

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: *r*^{2} = 0.9992$, and the mean square error is 0.0126 radians or 0.72 degrees. So we can write , in radians. In degrees this is .

For example, say you have an angle with — the smaller angle of a 5-12-13 right triangle. Then we get . In fact — 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

where . 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

Applying this to our test angle with , we get radians, while in truth .

This formula is so nice that I can’t help but suspect there’s a simple derivation. Any takers?

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I assume you mean and .

Here’s one way you could invent that approximation: you recall that

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:

The choice makes the terms equal, and then rearranging yields

as desired.

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

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

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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