2 sin^2 + sin x - 3 = 0

x = 2 π n_1 + π/2 for n_1 element Z

or x = π + sin^ (-1) (3/2) + 2 π n_2 for n_2 element Z or x = 2 π n_3 - sin^ (-1) (3/2) for n_3 element Z

Step-by-step explanation:

Solve for x:

-3 + sin (x) + 2 sin^2 (x) = 0

The left hand side factors into a product with two terms:

(sin (x) - 1) (2 sin (x) + 3) = 0

Split into two equations:

sin (x) - 1 = 0 or 2 sin (x) + 3 = 0

Add 1 to both sides:

sin (x) = 1 or 2 sin (x) + 3 = 0

Take the inverse sine of both sides:

x = 2 π n_1 + π/2 for n_1 element Z

or 2 sin (x) + 3 = 0

Subtract 3 from both sides:

x = 2 π n_1 + π/2 for n_1 element Z

or 2 sin (x) = - 3

Divide both sides by 2:

x = 2 π n_1 + π/2 for n_1 element Z

or sin (x) = - 3/2

Take the inverse sine of both sides:

Answer: x = 2 π n_1 + π/2 for n_1 element Z

or x = π + sin^ (-1) (3/2) + 2 π n_2 for n_2 element Z or x = 2 π n_3 - sin^ (-1) (3/2) for n_3 element Z