Find the general solution of cos2x + 3 sin x =2

To find the general solution of the equation cos(2x) + 3sin(x) = 2, we can use trigonometric identities to simplify the equation and solve for x.

Step 1: Start by using the double angle identity for cosine:
cos(2x) = 1 - 2sin^2(x)

The equation becomes:
1 - 2sin^2(x) + 3sin(x) = 2

Step 2: Rearrange the equation and combine like terms:
2sin^2(x) - 3sin(x) + 1 = 0

Step 3: Factor the quadratic equation:
(2sin(x) - 1)(sin(x) - 1) = 0

Step 4: Solve each factor separately:
Setting the first factor to zero:
2sin(x) - 1 = 0
sin(x) = 1/2

Using the inverse sine function, we find the solutions for sin(x) = 1/2:
x = π/6 + 2πn or x = 5π/6 + 2πn, where n is an integer.

Setting the second factor to zero:
sin(x) - 1 = 0
sin(x) = 1

Using the inverse sine function, we find the solution for sin(x) = 1:
x = π/2 + 2πn, where n is an integer.

Therefore, the general solution for the equation cos(2x) + 3sin(x) = 2 is:
x = π/6 + 2πn, 5π/6 + 2πn, π/2 + 2πn

Where n is an integer, allowing for an infinite number of solutions.