There are many ways to approach this. In general the sine of an angle is easier to approximate for very small angles because:
#sinx ~= x# for #x ~=0#
where #x# is expressed in radiants, or more precisely using the MacLaurin series:
#sinx = sum_(n=0)^oo (-1)^nx^(2n+1)/((2n+1)!)#
and based on Lagrange's theorem on the rest:
#sinx = sum_(n=0)^N (-1)^nx^(2n+1)/((2n+1)!) + R#
where #absR < x^(2N+1)/((2N+1)!)#
So we can find the well-known sine value closest to #53°#, use the formula for the sine of the sum of two angles, and approximate the sine of the difference, for example:
#sin(53°) = sin(60° -7°) = sin(60°)cos(7°) -cos(60°)sin(7°)#
Now:
#sin(60°) = sqrt3/2 ~= 0.86602540378#
#cos(60°) = 1/2 =0.5#
and in radiants #7° = (7pi)/180#
Then:
#sin((7pi)/180) = (7pi)/180 -1/6((7pi)/180)^3 ~= 0.121869#
with an error #R < 2.26827545e-7# that is to the sixth decimal place.
and:
#cos((7pi)/180) = sqrt(1-sin^2((7pi)/180) ) ~= 0.992546#
So:
#sin(53°) ~= 0.866025 * 0.992546 - 0.5* 0.121869 = 0.798635#
