sin(x) = \(x – x^3/3! + x^5/5! – x^7/7! + x^9/9! – \)…
The sin(x) is plotted in black.
Lets break down the amount of work we need to do depending on size of the angle.
2 Terms are needed for angles 0 through .9 radians
3 Terms are needed for angles >.9 and <1.57
4 Terms are needed for angles >1.57 and <2.5 radians
5 Terms are needed for angles >2.5 and <3.3 radians
6 Terms are needed for angles >3.3 and <4 radians
7 Terms are needed for angles >4 and <4.5 radians
8 Terms are needed for angles >4.5 and <5.5 radians
9 Terms are needed for angles >5.5 and <=2*pi radians
This can be A LOT of WORK with pencil and paper.
Fortunately, the sin of any (A)NGLE > 2*pi will be identical to the sin of an (a)ngle less than or equal to 2*pi.
So for example, given an (A)ngle of 31.72 radians its corresponding (a)ngle will be .304
n = int(A/(2*pi))
a = A-2*n*pi
Given A = 31.72 then n = 5
hence a = 31.72-(2*5*pi)= .304
GREAT!! we only need 2 Terms x – x^3/fact(3)
This interactive Math Dynamics Pallet demonstrates how the Maclaurin Expansion Series can be used to calculate the sin of any angle