Sunday 11 December 2022

An 18th Century quick Approximation for Angles in Right Triangle (Repost)

The triangle above is a right triangle, and almost no one who reads this blog didn't know that. In fact for most people who have had a pretty good math background, you would be surprised to read a property or theorem about right triangles that you didn't know, or at least hadn't heard, especially if it dates back to the 18th Century.

Ok, so if I give you the three side lengths of a right triangle, how would you find the smallest angle..... Hold on, No calculator, and No tables...
I had no idea how to achieve such a result when I came across such a method by a Hugh Worthington in "An essay on the Resolution of Plain Triangles, by Common Arithmetic." It was in an anthology of math writing from the 1500's to the present in "A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing" by Benjamin Wardhaugh (pg 97), credited to Hugh Worthington in 1780.

So How is it done? Well, in the words of the author, "half the longer of the two legs added to the hypotenuse, is always in proportion to 86 as the shorter leg is to its opposite angle. "

As a math equation we with c as the hypotenuse and a as the smallest leg, he states that \(\frac{b/2+c}{86}=\frac{a}{A}\)

In the common 3,4,5 triangle that produces \(\frac{7}{86}=\frac{3}{A}\) and A would be equal to 36.85714286 (Ok, I divided it out on my calculator). Then I checked using Arcsin(3/5) and got 36.86989765...... OK, That seems to be a really good approximation, and I checked with a much smaller angle (the 7, 24, 25 triangle) and it was also very close.

I've been working on this for a while and can't come up with how he might have arrived at this approximation, nor can I find any other use of this. Would love to know if it was explained somewhere, or other works using it.

Addendum: Some really good math in the comments leading to the idea from Matt McIrvin and Paul Hertzer that "I think the leading error in this rule actually comes from the use of 86 degrees as an approximation to 1.5 radians." So using 85.94 or so as the constant improves the already awesome (my view) approximation accuracy. Thanks guys.. now can anybody find another publication using this idea?

Shortly after I wrote this, John Golden, an amazing teacher of math educators created a geogebra demo to greatly improve this, so I immediately stole it and added it here. Thanks John.

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