Trig functions.... waht exactly is the function?

Factual Questions
1

Well, if i type sin(sqrt(3)/2) in my calculator, i know what im doing, but i have no idea what the calculator is doing… every since i first began “using” trig functions, ive been wanting to know what the exact fuctions, but ive never seen how its calculated.

Ok, so to rephrase my question, what the hell does my calculator do when i enter sin(60)??

2

Take a triangle with a 90 degree angle and a 60 degree angle. The side opposite the 90 degree angle is the hypotenuse, the side opposite the 60 degree angle is called the opposite side. The sine of the 60 degree angle is equal to the length of the opposite divided by the length of the hypotenuse. This will be the same amount no matter what size the triangle is.

3

I didn’t really finish answering that, I think the calculator has that memorized.

4

I believe modern calculators use look-up tables for most trig functions, and interpolate as necessary.

The alternative would be a Taylor series expansion. From a mathematical point of view, definitions of trig functions in terms of Taylor series have the advantage of allowing complex variables.

5

Why would modern calculators interpolate, since calculators going back to the HP-35 (circa 1974) did these functions with Taylor series? The algorithms are long since done, and even a 30 year old model is plenty fast enough.

6

Just so I can have some input:

My calcuator (TI-89) gives a different result for sin(x) and sin(x+ .00005). That would be a huge lookup up table, and I don’t know how much harder interpolation is compared to a taylor series.

7

Previous thread discussing this:

http://boards.straightdope.com/sdmb/showthread.php?threadid=146550

8

The formula used in Bill Gate’s first version of Basic was:

sin(x) = x - x[sup]3[/sup]/3! + x[sup]5[/sup]/5! - x[sup]7[/sup]/7! + x[sup]9[/sup]/9!

(MacLaurin series)

cos and tan were derived from sin.

Monte Davidoff wrote the math bits of the program, including the above. Bill himself wrote most of the rest. Paul Allen did the remaining bits.

9

Some interesting background on the source code for that version of Basic can be found here: Raiders of the Lost Altair BASIC Source Code

Reuben Harris is a poster on SDMB, under the username Reuben. Reuben’s disassembly efforts can be seen here: Altair BASIC 3.2 (4K) - Annotated Disassembly

10

Not even the previously linked thread offered this explanation of TI calculators’ internal trig computation algorithms.