1-3cos^2x = -5sin x

-3cos^x + 5sin x +1 = 0

-3(1-sin^2 x) + 5sin x +1 = 0

3sin^2 x + 5sin x -2 = 0

then x = sin x

replace sine with x so it will give you a quadratic equation.

3x^2 + 5 x -2=0

solutions are

x = -2/3 and x = -1

then your x was sine, so

sinx = -2/3 , sinx = -1

then if 0<(or =)x<(or=) 2pi

x=5.665 x = 4.712