Sin (2x -9)= cos (x +30) to solve for x

A graphing calculator shows four (4) zeros of
  f(x) = sin(2x -9°) -cos(x +30°)
in the range 0° ≤ x ≤ 360°

Solutions are x ∈ {23°, 129°, 143°, 263°).

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You can make use of a couple of trig identities to rearrange this equation.
  cos(x) = sin(x+90°)
  sin(a) -sin(b) = 2*cos((a+b)/2)*sin((a-b)/2)
So
  sin(2x -9°) -sin(x +120°) = 2*cos((3x +111°)/2)*sin((x -129°)/2) = 0
The cosine factor will be zero for
  (3x +111°)/2 = n*180° +90°
  3x -69° = n*360°
  x = 23° +n*120° . . . . . . for any integer n
The sine factor will be zero for
  (x -129°)/2 = n*180°
  x = 129° +n*360° . . . . . for any integer n

Combined, these solutions give the ones listed above in the range 0..360°.