Question

Find all the values of x between 0 and 2(pi) for which
sin x + (root 3)cos x = 0

Answer 1

Whenever you see sines and cosines and sqrt 3, draw a thirty, sixty ninety triangle.
the sin of the 60 degree angle is (sqrt 3)/2
the cosine of the 60 degree angle is 1/2
sqrt(3)/2 = (sqrt 3)(1/2)
Now the quadrants where the sine and cosine have opposite signs are quadrant 2 and quadrant four. So look in those two quadrants.

Answer 2

another not necessarily better approach would be ...

sinx = -√3cosx
tanx = -√3

from the same triangle as Damon's we would get a "basic" angle of 60º or pi/3 radians

but the tan is negative in II or IV, so
x = pi - pi/3 or 2pi - pi/3
or x = 2pi/3 or 5pi/3