# Optimal 5th Degree Polynomial which Approximates the Sine Function

*Describes a 5th-degree polynomial that was found to approximate the sine function precisely up to 4 decimal digits. Useful for resource-constrained devices, games, or any applications where the precision of the least significant digits is not important.*

Our goal is to aproximate the sine function `sin`

between 0 and `pi/2`

.

We found the best coefficients for a polynomial of degree 5 in the form `f(x) = c5 x^5 + c3 x^3 + c1 x`

to be the following:

c1 = 0.99977140751539240 c3 = -0.16582704279148017 c5 = 0.0075742477643552034

According to Wolfram Alpha, the largest error in the domain between 0 and `pi/2`

is of magnitude ~0.00016 at x~=1.5708, so this function should be good to use for up to 4 decimal digits.

Here's the query:

maximize abs(0.0075742477643552034 x^5 + -0.16582704279148017 x^3 + 0.99977140751539240 x - sin(x)) with x between 0 and pi/2

The coefficients were found by searching in `R^3`

for the optimal vector of coefficients `(c5, c3, c1)`

for the function `f(x) = c5 x^5 + c3 x^3 + c1 x`

which minimize the RMSE (Root Mean Squared Error) between `f(x)`

and `sin(x)`

, assuming `x`

in interval between 0 and `pi/2`

.