sin2x = (radical 3)/ 2

Question

solve each equation on the interval [0,2pi)

Answer 1

among the repertoire of "trig things I should know" has to be the trig ratios of standard angles like 0, 30, 45, 60, and 90°
one is that sin 60° = +√3/2
as well as sin 120° = √3/2 , by the CAST rule

so if sin 2x = √3/2, then
2x = 60° or 2x = 120°
x = 30° or x = 60°

secondly, since the period of sin 2x = 180°, adding 180° to any solution yields another solution , so
x = 30+180 or 210°
x = 60+180° = 240°

in radians that would be
x = π/6 , π/3 , 7π/6 and 4π/3

Answer 2

so what about
tan3x= radical3 / 3

Answer 3

I would do it the same way, except √3/3 is not associated with one of the standard angles.
did you mean √3/2 again ?

so if sin 3x = √3/2
3x = 60° or 3x = 120°
x = 20° or x = 40°

period = 360/3 = 120°
so add 120° to any angle until you go over 360°
convert all answers to radians.
e.g. 20° = π/9 radians etc