Solve for x:
sin(2 x) - sqrt(3) cos(x) = 0
Expand trigonometric functions:
2 cos(x) sin(x) - sqrt(3) cos(x) = 0
Collecting terms, 2 cos(x) sin(x) - sqrt(3) cos(x) = (2 sin(x) - sqrt(3)) cos(x):
cos(x) (2 sin(x) - sqrt(3)) = 0
Split into two equations:
cos(x) = 0 or 2 sin(x) - sqrt(3) = 0
Take the inverse cosine of both sides:
x = π/2 + π n_1 for n_1 element Z
or 2 sin(x) - sqrt(3) = 0
Add sqrt(3) to both sides:
x = π/2 + π n_1 for n_1 element Z
or 2 sin(x) = sqrt(3)
Divide both sides by 2:
x = π/2 + π n_1 for n_1 element Z
or sin(x) = sqrt(3)/2
Take the inverse sine of both sides:
x = π/2 + π n_1 for n_1 element Z
or x = π - sin^(-1)(sqrt(3)/2) + 2 π n_2 for n_2 element Z or x = sin^(-1)(sqrt(3)/2) + 2 π n_3 for n_3 element Z
sin(2 x) - sqrt(3) cos(x) ⇒ sin(2 (π/2 + π n_1)) - sqrt(3) cos(π/2 + π n_1) = 0:
So this solution is correct
sin(2 x) - sqrt(3) cos(x) ⇒ sin(2 (329 π - sin^(-1)(sqrt(3)/2))) - sqrt(3) cos(329 π - sin^(-1)(sqrt(3)/2)) = 0:
So this solution is incorrect
sin(2 x) - sqrt(3) cos(x) ⇒ sin(2 (184 π + sin^(-1)(sqrt(3)/2))) - sqrt(3) cos(184 π + sin^(-1)(sqrt(3)/2)) = 0:
So this solution is incorrect
The solution is:
Answer: | x = π/2 + π n_1 for n_1 element Z or x = π - sin^(-1)(sqrt(3)/2) + 2 π n_2 for n_2 element Z or x = sin^(-1)(sqrt(3)/2) + 2 π n_3 for n_3 element Z
