Find the values of theta in the interval 0°<x<180° for which: 4root3sin(3x+20°) = 4cos(3x+20°)?

I assume by theta, you mean x. Lets convert those degrees to radians first: 20degrees -> pi / 9 radians 180degrees -> pi radians 4(3^(1/2))sin(3x + pi/9) = 4cos(3x + pi/9) -> divide both by cos(3x + pi/9) 4(3^(1/2))tan(3x + pi/9) = 4 -> divide both by 4 (3^(1/2))tan(3x + pi/9) = 1 -> divide by root 3 tan(3x + pi/9) = 1 / (3^(1/2)) -> take inverse tan of both 3x + pi/9 = tan-1(1 / (3^(1/2))) -> simplify 3x + pi/9 = pi/6 + pi*n-> subtract pi/9 from both sides 3x = pi / 18 + pi*n -> divide by 3 x = pi / 54 + (pi*n / 3) Now if we substitute values of n, as long as the result of x is in the domain 0 < x < pi, then that is a valid solution. if n = 0, x = pi/54 if n = 1, x = pi/54 + pi/3 if n = 2, x = pi/54 + 2pi/3 Those are the only three possible solutions, all others are outside of the domain.