Final answer:
To solve the trigonometric equation, rewrite it as a standard quadratic equation in terms of sex, use the quadratic formula to find y, and then find x by taking the arcsine of each solution for y.
Step-by-step explanation:
To solve the equation 3sin^2x - 5sinx - 2 = 0, we can treat it as a quadratic equation in terms of six. Let's set y = sinx. The equation then becomes 3y^2 - 5y - 2 = 0.
This is a standard quadratic equation, and we can solve for y using the quadratic formula, y = (-b ± √(b^2 - 4ac))/(2a). After finding the values of y, we can solve for x by taking sin^{-1}(y) for each solution of y that lies within the range of the sine function, which is [-1, 1].
Once we have the solutions for x, we need to consider the domain of x in the context of the problem, which is typically the set of all angles where the quadratic equation in six holds.