(Solution) - To accurately approximate sin x and cos x for inclusion -(2025 Original AI-Free Solution)

Paper Details

To accurately approximate sin x and cos x for inclusion in a mathematical library, we first restrict their domains. Given a real number x, divide by ? to obtain the relation
|x| = M? + s, where M is an integer and |s| ? ?/2.
a. Show that sin x = sgn(x) · (?1)M · sin s.
b. Construct a rational approximation to sin s using n=m=4. Estimate the error when 0 ? |s| ? ?/2.
c. Design and implementation of sin x using parts (a) and (b)
d. Repeat part (c) for cos x using the fact that cos x = sin(x + ?/2).