Trigonometry and Making Stuff

September 21, 2006 | General | By: Mark VandeWettering

I must admit: I’m a bit of a math nut. Solving interesting mathematical problems makes me feel cool. But I must admit that I’ve suspected for a long time that one branch of mathematics is almost completely useless, and that’s trigonometry.

By way of evidence, witness the following link:

Per Vivere [To Live]: Trigonometry and Making Stuff

It’s very nearly the perfect example of why trigonmetry is useless. Witness the six stage derivation, which includes no less than 12 trigonometric functions, before arriving at the final answer, that the angle required is 16.26°. Then, at the end of it all, what does he do? He needs to set his protractor, which he doesn’t trust, so he computes the tangent of 16.26°, multiplies it by a baseline of 300mm, and arrives at 87.498837mm.

But what do you do if you don’t have a scientific calculator? Or even a protractor? If you look at the problem, you can easily reduce it to a simple application of the Pythagorean theorem (you all remember, the a2 + b2 = c2 thing?). From his drawing, it is apparent that the block that you want to cut has a hypoteneuse of 125, and has one leg with length 35. That means that the other leg is the sqrt(1252-352), or 120mm. To set his angle gauge over a 300mm distance, (2.5 times the baseline he actually has), he needs to multiply the thickness of his piece by the same amount, and arrives at the precise answer: 87.5mm.

Sines, cosines and tangents all arise from the properties of right triangles. In virtually every case, it is simpler to draw a diagram and work out the resulting lengths via the Pythagorean theorem. The only time you need to remember what a sine, cosine or tangent is are in those rare situations where you actually need to know what an angle actually is, and even then, it’s best to just do that conversion from two appropriate lengths at the end of your problem.

Of course carpenter’s have known this for a long time, that’s why you don’t see any degree measures on a framing square, although their use is fading into the past as well. Too bad.

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Addendum: Wikipedia has a brief introduction to the use of the steel square.