Is it possible to find an exact answer for something like sine(46 degrees)? I understand how to use the sum formula for sine(a+b) to get something like sine(75). I also see how you could use the sine(a/2) formula to get something like sine (22.5) and keep halving it to get lower and lower.
But if one only knows exact values for the sine of 30,45, and 60, the sum formula and the halving formula, wouldn't it be impossible to find an exact answer for the sine(23)?
This is exactly the reason why Taylor Polynomials were created. If you think about a calculator, all that it can do is add (and by extension, subtract, multiply, divide and exponentiate). So when programming a calculator to evaluate (say) the sine of any angle, it needs to be programmed with some kind of polynomial (in other words, some combination of addition, subtraction, multiplication, division and exponentiation). I suggest you google "Taylor Series" for more information.
Calculators do not in general use Taylor series representations of special functions, they use polynomial approximation over some range (which are not Taylor polynomials for the function but something like a least absolute error approximation on the range) and reduction formulae to reduce an argument to that range.
You can also Google for CORDIC algorithms which is another approach.
CB