1-sin(x)=cos(x)√3,
½-½sin(x)=(√3/2)cos(x).
cos(π/3)=½, sin(π/3)=√3/2, and sin(π/6)=½, so:
sin(π/6)=(√3/2)cos(x)+½sin(x)=sin(π/3)cos(x)+cos(π/3)sin(x)=sin(x+π/3).
Therefore π/6=x+π/3, x=-π/6. Also since sin(π/6)=sin(5π/6):
5π/6=x+π/3, x=5π/6-π/3=π/2.
More generally: π/6+2πn=x+π/3 and 5π/6+2πn=x+π/3, where n is an integer, so:
x=2πn-π/6=(12n-1)π/6 and x=2πn+π/2=(4n+1)π/2.
x is in radians. To convert to degrees, replace π by 180°.