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°.
