How Did Trigonometry Functions Originate and Evolve?

[DM]

How would you, to the best of your abilities, explain to someone like me how trigonometry functions emerged? I'm not in any way or form an advanced mathematics student and in this specific subject I only know the basics of the very basics (i.e how to find the sides and angles of right triangles). I should also inform you that I have already researched in wikipedia and many other web sources. To my discontent appreciation, none of the websites have presented me with straight forward answers. Hence why I choose to finally succumb and consult the professionals of this magnificent website.

Not only do I very politely beg you to provide me with web links that concerns about the history of trigonometry functions but also, and more importantly should I mention, how the values of each individual trigonometry function was devised.

Thanking you in advance

 

[Integral]

Since this is more of a math history question then a Math question, I have moved the thread to History.

http://www.cartage.org.lb/en/themes/Sciences/Mathematics/Trigonometry/history/History%20.html"

 

[DM]

Thank you very much Integral. I do apologise for posting in the wrong section of the forum.

Google lessons? I believe I was somewhat unfortunate in not discovering the website that you've just posted. Perhaps I did indeed write the wrong search words on the search engine.

 

[selfAdjoint]

The ancient Greek geometers discovered the first trig function, the chord; this was a fairly obvious induction from working with circles; the ratio of the chord subtended by a given angle at the center of a circle to the circle's radius is seen to be a well-defined property of the angle.

These geometers were more interested in spherical geometry than in plane, because of their interest in models of planetary astronomy. They discovered a "law of chords" in spherical geometry, parallel to the "law of sines" in modern spherical trig, and with this they were able to solve spherical triangles, although with more work than we would have today. The math in Prolemy's Almagest is a lot of this. Ptolemy also includes a table of chords and shows you how to calculate one for yourself.

In some way that historians are not sure of, the Greek work was apparently transmitted, though not completely, to India. The early Indiaqn astronomers introduced the sine to replace the chord. This is also easy to see in circular geometry: the sine of an angle at the center of a circle is the half chord (in the Greek sense) of twice the angle. This enables you to use Pythagoras' rule to define the cosine. Indian astronomers apparently discovered the law of sines and used it in their own astronomical calculations; The names Aryabhatta and Bhaskara are notable here. The name given to the sine in India was j'ya (approximate spelling) which just means "chord".

The Islamic astronomers picked up the Indian sine function, calling it by the Sanscrit name j'ya. And when the Western scholars translated the Islamic works, they mistranslated j'ya which they thought was the Arabic word for a gulf or bay, a sinus which means gulf or bay in Latin.

 

[DM]

Very interesting indeed. I thank you for putting time into the writting.

 

[selfAdjoint]

Rereading it I see that I may have given a misapprehension. Until modern times everybody thought of the trig functions as lengths, not ratios. So the chords and sines came out as multiples of the radius of whatever circle they were defined in. It is odd that though everybody from the Greeks on was persectly aware of the properties of similar triangles, they never realized they could use that in defining their trig function. And of course the whole idea of a "function" is modern too. None of that mattered when they got down to doing calculations, though.

 

[BobG]

How would you, to the best of your abilities, explain to someone like me how trigonometry functions emerged? I'm not in any way or form an advanced mathematics student and in this specific subject I only know the basics of the very basics (i.e how to find the sides and angles of right triangles). I should also inform you that I have already researched in wikipedia and many other web sources. To my discontent appreciation, none of the websites have presented me with straight forward answers. Hence why I choose to finally succumb and consult the professionals of this magnificent website.

Not only do I very politely beg you to provide me with web links that concerns about the history of trigonometry functions but also, and more importantly should I mention, how the values of each individual trigonometry function was devised.

Thanking you in advance

Trig functions led to some other important math functions, as well.

If you know your sum and difference theorems, you might realize that trig tables could be used to solve difficult multiplication problems. If you had the values for the sine and cosine of every angle from 0 to 90 degrees in a table, you could look up one of the numbers you wanted to multiply in the sine column, the other in the cosine column, and use the trig table to reduce your difficult multiplication problem into a simple addition and subtraction problem.

For example:

sin(A)*cos(B) = (1/2)sin(A+B) + (1/2)sin(A-B)

Thus to multiply 173.65*9.9027, you look up in tables and find
0.17365 = sin(10), 0.99027 = cos(8) and the above formula gives:

sin(10)*cos(8) = (1/2)(sin(18) + sin(2))

From tables, sin(18) = 0.30902 sin(2) = 0.03490

sin(18) + sin(2) = 0.34392 and

(1/2)(sin(18)+sin(2)) = 0.17196

Giving 0.17365*0.99027 = 0.17196

You have to mentally keep track of the fact that 173.65 equals .17365 x 10^3 and that 9.9027 equals .99027 x 10^1. You add your powers of ten and realize that your final answer has to be .17196 x 10^4, or 1719.6

Eventually, John Napier took this a step further and developed "logarithms". Same basic principle: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html. Napier's logarithms weren't to any base, but Henry Briggs worked with Napier to eventually develop base 10 logarithm of e and eventually Nicolaus Mercator developed a table of base e logarithms.

William Oughtred used the relationship of logarithms and numbers and used them to create a couple of sliding scales. Now instead of having to carry around a table of trig values or logarithms, you carry them on a couple of sticks that were rigged so that one stick could slide back and forth - the first slide rule, which was an extremely powerful calculating device that was popular for 340+ years - until the first "electronic slide rule" was developed.

Note: there's a difference between a calculator, which traditionally was used for mechanical devices that could add and subtract, and a slide rule, which could do much more advanced jobs, such as multiplication, subtraction, powers, roots, solve quadratic equations, solve complex number problems, etc. Seeing as how most people only need a 'calculator' capable of balancing a checkbook, the term 'calculator' has come to describe even the more advanced electronic calculating devices.