How did trigonometry come about?

[JyN]

I am a first year engineering student and i am starting to really understand/be very interested in math. I have started proving random theorems after being intrigued by things in class and am really appreciating things like the fundamental theorem of calculus.

I am very curios about how trigonometry is derived. How do u come to create a function like sine or cosine? I can see the importance of arc length and the pythagorean theorem, but i can't make the leap from there to being able to get the sine function, and to be able to determine the actual values of sin a, or cos b etc... Google searches havn't given me much either. Can anyone point me in the right direction?

EDIT: i should also be clear that i am not interested in the history, or the people that actually did this. I am interested in being able to start with very little, and know how to discover all of basic trigonometry as we know it today.

 

[Integral]

All that you ask for can be summed up in one word:

SOHCAHTOA

 

[Deveno]

well, the sine and cosine functions were not the first objects of consideration. originally, what was studied was the length of chords, the straight line segment connecting the endpoints of a circular arc. of course, if you bisect the chord by the x-axis, this is what would be seen nowadays as 2sin(θ/2). but back then, rene descartes has not yet seen the spider on the window-pane, so the idea of an "x-axis" didn't even exist.

also, as a hold-over from babylonian mathematics/astronomy, angle measure was in degrees, minutes (a minUTE part of a degree) and seconds (a second minutae, or minute part of a minute part).

the perpendicular line segment from the midpoint of the chord to the arc, also once had its own name: the versed sine (or "turned sine"). this was, at the time, the natural way to turn the triangle formed by the chord and two line-segments from the end-points of the arc through the center of the circle into 2 right-triangles, to which the theorem of pythagoras could then be applied.

if the circle was of unit length, this gives:

[chord(θ)]^2 = sin^2(θ) + versin^2(θ)

the supplantation of the versine function by the cosine function did not happen until much later (around 1600 i think).

the history of the word "sine" is kind of interesting. the hindus were the first to have a name for half-chords, which they called jya. the arabic mathematicians who first translated the indian works, transliterated this as "jiba" which later became corrupted to "jaib" meaning "fold". this was (most notably in fibonacci's liber abacus) translated into latin as "sinus", which became the recognized term for the half-chord.

early work in trigonometry depended heavily on tables of chords, which had to be painstakingly compiled. it was suspected that most values of chords were irrational, but the means to prove this did not exist until relatively recently (hence the problem of "squaring the circle" was not solved until the 1800's).

nowadays, with the almost universal teaching of coordinate systems, it is easy to see the sine as a y-coordinate on a circle, and the cosine as the x-coordinate. even so, computing the values of sin(a), for a given angle a, can be a daunting task. for certain angles (such as 30 degrees, 45 degrees, and 60 degrees), one can use elementary geometric arguments to find these sines and cosines. one can also has "double-angle" and "half-angle" formulas that allow for a broadening of these basic points. it was once a commmon practice to instruct students in the art of interpolating unknown trigonometric values from values published in a table, and every math book worth its salt, had such tables listed in the back.

the be-all and end-all of trigonometry in its modern form, is the fundamental identity:

sin^2(θ) + cos^2(θ) = 1.

this is just the the equation x^2 + y^2 = 1, of the circle of radius 1, which is also pythagoras' theorem in disguise.

for a proof of the "two-angle" formulas for sine and cosine, look here:

http://www.themathpage.com/atrig/sum-proof.htm

any proof of these relies on geometry (well, there is a version that uses complex numbers, but that is like going to law school to be able to prove stealing is wrong).

 

[Ben Niehoff]

One interesting fact is that it is possible to express

sin( 1 degree )

using a finite number of radicals (however, the actual expression would fill an entire page). It might be a fun challenge to figure out how to get it...you would learn some things about algebra along the way.

And if you figure that out, you might also see that it is possible to express

sin( 2 pi / 7 )

in radicals.