Answer:
To solve the equation sin 2x = 4cos 2x, we can use the trigonometric identity:
sin 2x = 2sin x cos x
cos 2x = cos^2 x - sin^2 x = 1 - 2sin^2 x
Substituting these identities into the equation, we get:
2sin x cos x = 4(1 - 2sin^2 x)
Expanding and simplifying, we get:
2sin x cos x = 4 - 8sin^2 x
Rearranging, we get:
8sin^2 x + 2sin x cos x - 4 = 0
We can factor the left-hand side of this equation:
(2sin x - 1)(4sin x + 4) = 0
Setting each factor equal to 0 and solving for sin x, we get:
2sin x - 1 = 0 or 4sin x + 4 = 0
sin x = 1/2 or sin x = -1
If sin x = 1/2, then x is a multiple of π/6, so the general solution in this case is:
x = π/6 + 2πn or x = 5π/6 + 2πn
Where n is an integer.
If sin x = -1, then x is a multiple of π plus an odd multiple of π/2, so the general solution in this case is:
x = π/2 + πn
Where n is an integer.
Therefore, the general solution for sin 2x = 4cos 2x is:
x = π/6 + 2πn or x = 5π/6 + 2πn or x = π/2 + πn
Where n is an integer.