How did we create the trigonometric functions?dcc

[PhotonW/mass]

how was the trigonometric functions created? how did mathematicians find cosine, sine, tangent, etc. without a calculator. basically how would i find the trigonometric functions after the collapse of civilization and it was up to me to rewrite all the charts and program all the calculators that finds all the trigonometric functions? sorry for the bad grammar. i am using tablet.

 

[caveman1917]

Look up "Taylor series".

 

[HallsofIvy]

The original tables of trig functions, ante-dating the invention of Taylors series, were made by actually drawing very large right triangles and measuring the sides very accurately.

 

[The_Duck]

I believe another technique was to start from simple angles like 45 degrees and 30 degrees for which the values are trivial, and then repeatedly apply the half-angle formulas and the angle sum and difference formulas to get many other angles.

 

[FactChecker]

Famous mathematicians of old used to hire "idiot savants" who could make those tables in their head. I don't know if the savants could even explain how they were doing those calculations. I believe that Gauss had that ability in addition to his other genius capabilities.

 

[ellipsis]

If you have a bunch of neurons clumped together, can you train that system to accurately render a triangle, and take measurements of the sides? (or whatever specific operation needs to be done to calculate them). Something odd is going on if idiot savants can do these calculations and we don't know how.

The Duck: That's how you get the values of trigonometric functions in terms of square roots. You cannot do that to find the exact value of "mean" expressions though, like sin(e).

 

[FactChecker]

Something odd is going on if idiot savants can do these calculations and we don't know how.

It certainly is very strange. When they are asked how they are doing their calculations, some of them talk about the smell, color, feel. sound, etc. of numbers. It's sort of gibberish. It's as though parts of their brain that are usually used for other functions (small and color) are being put to use in the calculations.

But maybe I have departed from the question that the OP had in mind. I thought that the question was how people could make accurate tables of those functions before computers. They always had logical ways, but the calculations were very tedious. So they gave the job to savants.

 

[256bits]

how was the trigonometric functions created? how did mathematicians find cosine, sine, tangent, etc. without a calculator. basically how would i find the trigonometric functions after the collapse of civilization and it was up to me to rewrite all the charts and program all the calculators that finds all the trigonometric functions? sorry for the bad grammar. i am using tablet.

Well, you would certainly have a problem, wouldn't you.

Have a few good sharp pencils, a straight edge, and some kind of linear measuring stick, a divider, and a lot of paper, and a good grasp of geometry and trigonometry.

If you can find anyone of the values for the trigonometric function, such as the sine of an angle, then it is relatively trivial to find all the familiar others, such as cosine, tangent, secant, cosecant, and cotangent, as they are are related by simple formula.

This picture has some of the others that you most likely have not heard about, and their relationships with a circle, which you could add to your chart.

http://en.wikipedia.org/wiki/Versine
Quote:
Historically, the versed sine was considered one of the most important trigonometric functions,[2][3][4] but it has fallen from popularity in modern times due to the availability of https://www.physicsforums.com/wiki/Computer and scientific https://www.physicsforums.com/wiki/Calculator .
Unquote.

Here is a picture of common angles and their sine, cosine.

Here is a picture of angle sum-difference with a relationship to a rectangle.

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

One of the first tables goes back to Ptolemy, and it lists chord lengths and not the sin, cos or those we are familiar with
http://en.wikipedia.org/wiki/Ptolemy's_table_of_chords

First problem you face, if you want to start with a circle, is dividing you circle up into equal angled segments.
Whether you want to continue with what we call degrees, a degree being 1/360 of the whole angle of a circle or something else is up to you, but since you have only a straight edge, dividers and ruler to measure things, and no calculator, some choices might be easier than others so you don't get as many of those nasty decimal places
You will notice that an equalateral triangle has angles of 60 degrees at each corner, so that is one place to start, and you could divide your 60 degrees successfully to get the 30, 15, 7.5, ... Try to get 5 degrees or 7 degree angle. Hmm. Interesting. Better call Ptolemy how did he do that.
Also 6 equalateral triangles fit inside a circle so that is just neat.