cos(x)+sin(x)=⅓; sin(a+b)=sin(a)cos(x)+cos(a)sin(x).
If a=π/4, sin(a)=cos(π/4)=√2/2 so
sin(π/4+x)=(√2/2)(cos(x)+sin(x))=√2/6,
π/4+x=sin-1(√2/6), x=sin-1(√2/6)-π/4=0.2379-0.7854=-0.5475 radians approx.
Other values are found by adding 2πn (for integer n). For example, n=1, x=5.7357 radians.
sin(5π/4)=cos(5π/4)=-√2/2, so sin(5π/4+x)=-√2/6, x=-0.2379-3.9270=-4.1649 radians approx.
Other values are -4.1649+2πn, for example, n=1, and x=2.1183.