I'm interested in learning the methodology to calculate trigonometric functions without a calculator. I was told by a college chum that the CRC Handbook of Mathematical Tables would include the methodologies to calculate these functions. I listed every single function, law, rule, etc. I could possibly imagine and more, it listed table after tables of calculated trig functions, but it did not show me HOW to calculate them myself.
I admit as an engineering major I do not like to have a subject waved in front of my face and then told..."To calculate sine, press sine x and then..." I about went batty in my lecture course when they tried to explain away sine, cosine, tangent, etc. by means of "pressing the sine button". If I wanted to be taught how to use a calculator I could simply read the instruction manual. What I want to know is HOW to calculate it. Not how to press a bunch of buttons and let a machine calculate it for me. I see this as a great failure in our math education programs around the US...We don't explain and prove these sorts of things and wonder why students don't understand.
Simply put, I understand there are some Taylor series I will learn in calculus that can be used to find sine, cosine, tangent, etc. however, if that was the only way to calculate sine, cosine, tangent, etc. I suspect mathematics did a lot of magic to get where it is today. I discovered through a variety of sources that one can calculate this based on double circles, where one is within another. Secondarily, I've looked at the idea of calculating sine based on chords, where sine of an angle theta is half the chord of twice the angle.
If this is a possible method, then I need a methodology to calculate the chords. If I am given an angle, I am curious how I might use the chord to calculate that angle. I understand that Hipparchus used an arch length of 1, still not sure how he figured the ratio properly, but by using an arc length of 1 he was able to calculate subsequently degrees of an angle. Also Ptolemy's theorem is another possible explanation from what I've been reading. However, no where can I find a quality source to explain this problem.
I can't believe in the 21st century that calculating trigonometric functions could be so difficult. I'm feeling rather like I was deposited in the dark ages of mathematics, where one group is passed needing to know how to calculate it and are fine with letting a calculator solve the problem, or prefer to use calculus to explain things, or who simply don't have a clue about math in general. There should be a relatively rigorous method for calculating these functions. If there wasn't, how on earth did we discover modern mathematics? During the period of Calculus, Descartes, etc. there had to have been a methodology to calculate these functions.
Can someone please enlighten me so I am not so lost?