I have been completely spoiled by calculators and computers. However, it has been so long that I don’t remember if I EVER knew how to do this. But at some point somebody had to be able to do it.
Problem:
Without using Trig or Log tables or a scientific calculator or a computer, and given one leg and one angle or two legs of a right triangle, how would I calculate the other legs and angles of the triangle?
I already know things like Opposite/Hypotenuse = sin(Angle), but how do I convert the sine into its angle? Or a cosine or tangent?
Life was so much easier when I only had to push a button.
Aside from a few simple cases (cosine of 60 degrees = 1/2, cosine of 45 degrees = 1/square root of 2)the sines and cosines are nontrivial functions of the angles. If you can’t use a calculator or a table, you’re forced to calculate via trigonometric relationships or through the use of series. Series solutions will always work, and you can do them by hand, but you’re gonna wish you had a calculator. Or at least an adding machine.The following give the sine or cosine in terms of radians, not degrees. There are pi radians in 180 degrees, making a radian about 57 degrees.
sin(x) = x -((x^3)/3!)+((x^5)/5!)((x^7)/7!)+…
cos(x) = 1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!)+…
arcsin(x) = x + (x^3/6)+(13)x^5/((24)5)+…
In the Above 5! = 54321, etc.
In the days before calculators, scientists and engineers gor sines, cosines, and tangents accurate to three places from their slide rules, using the S, T, and ST scales. I’m the last generation that had to do this, I think – I used a slide rule for my first two years at MIT, switching to a calculator when the HP-25 came out at under $200.
As I sat with a calculator earlier, I was beginning to figure that out. And, by the looks of your formulae, I guess I NEVER knew how to do this. And it doesn’t help that I was never a good, much less great, math student and it has been 25 years since my last math class. If we replace every cell every seven years, that was nearly four brains ago. My current brain is better with math, though. Or more willing to try.
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Slide rule! Eeek! On Antiques Roadshow last night they had one from the 1890s that was cylindrical so you could pack EVEN MORE functions on it.
Fortunately for me and my pencil and paper, what I wanted to know this for was to write a triangle solution program for this TRS-80 Level 1 BASIC emulator in a browser. This was an extremely limited version of BASIC from the first TRS-80 computers without any math functions except ±*/ and exponents. (The square root function didn’t come along until Level 2, so that part will be fun.)
Update: EEEEEK! He didn’t emulate exponentiation! It was supposed to be available in Level 1! Good thing I have 16k to work in.
They have one of these on display in the Mathematics Undergraduate Department office at MIT. There’s another on display under the rotunda in the Capitol building in Washington D.C. (It was used in designing the dome). I’ve seen one or two others. How much was it on the Antiques Roadshow? I’ll bet it was worth a mint.
In the book The Great Escape by Paul Brickhill they talk about measuring the distance to the trees outside the camp by “rough trig”. They measured the tunnel length directly, with string. They came up short, you’ll recall. My conclusion: calculating trig functions by hand is hard.
How much precision do you need? If you only want the angles to a degree or two, your best bet is probably to make up a look-up table for the program to use. This could probably be done with an array.
“Only” $900 (compared with the $75k Charles Russell-original place card autographed by the artist and Will Rogers). If I heard right it was missing a magnifier, although it didn’t look like it needed one. At least, no more than your typical K&E slipstick. Less, really. The lines were wider.
Sorry, that smacks of work for me. I have a computer for that.
