Does anyone have a really good proof of the Pythagorean theorem? Just like absolute values, it was always (apparently) deemed to trival for any math course I took (my B.S. is in Physics). I’ve come up with my own proof, it involve using 4 triangles to construct a larger square. It relies on a triangle having 180 degrees and the area of a triangle being 1/2 height times width. There must be a delightfully brief and elegant proof.
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How are basic trig ratios (COS, SIN, TAN) calculated? When I asked my 7th grade math teacher how they were calculated, she said people just took protractors and drew them out on paper (honestly) although I find myself extremely cynical of this answer.
“The first thing a man will do for his ideals is lie.”
–Joseph A. Schempeter
threemae, the sin and cos functions are calculated as summations of infinite series. The exact formulas were given (I think by lynne) in a thread that I’m unable to locate at this time. Beyond that, it’s tan = sin/cos, csc = 1/sin, sec = 1/cos, cot = cos/sin.
IIRC they are calculaded via an infinite series, I’m not sure how anyone ever figured out what the infinite series should be.
(Caution: The mnemonic that follows may offend the politically correct.)
The simple trig functions are ratios of the sides of a right triangle.
An old way to remember them is “Big Chief Soh-Cah-Toa”.
In a right triangle, the functions for one of the non-right angles are:
Sine=opposite/hypotenuse
Cos =adjacent/hypotenuse
Tan =opposite/adjacent
Originally they were probably “calculated” by direct measurement, and the ancient Greeks knew ways to approximate them, but they weren’t happy with the ideas of irrational numbers or infinite calculations, so didn’t take it any further.
(Here’s hoping those bold characters work.)
Bob the Random Expert
“If we don’t have the answer, we’ll make one up.”
When I took trig 25 in a local technical college 25 years ago, our professor told us the mnemonic he used to remember the basic trig functions:
Sally Can Tell Oscar Has A Hard On Always.
Must have worked, I still remember.
Ugly
There are many proofs of the theorem. A book published in 1940 listed 367 of them. When President Garfield was a US Representative, he developed a proof (although it turns out it was a simple variant of an already well-known proof).
RJKUgly (does that make me rjkHandsome???), your mnemonic came from college. Mine was from grade school, IIRC. Even if it was in high school, it was in a Saskatchewan town even more than 25 years ago.
We missed all the good ones.
Bob the Random Expert
“If we don’t have the answer, we’ll make one up.”
The mnemonic I learned in 9th grade:
Oh Hell, Another Hour Of Algebra
Not as useful as some others, but more fun to say in front of a math teacher.