14 hours ago
- The sine of x radians and sine of x degrees are equal when x = 0 or x = 180π/(180 + π).
- Solutions to sin(x) = sin(πx/180) come from two families: x = 360πk/(180 − π) and x = 180π(2k + 1)/(180 + π) for integer k.
- The condition arises from angles having the same sine if they differ by a multiple of 2π or are supplementary.
- An alternative derivation uses the sine difference identity sin(A) − sin(B) = 2 cos((A + B)/2) sin((A − B)/2).
- Both solution families are dense in [0, 2π] modulo 2π, allowing approximation of any y in [−1, 1] by sin(x) = sin(x°).
- Examples show finding specific x values to approximate given y with small tolerance using large integer k.