Final answer:
To determine the number of solutions for sin²x + sinx - 6 = 0, we treat it as a quadratic equation with substitution, factor or apply quadratic formula, and then check for values within the range of the sine function.
Step-by-step explanation:
The question asks about the number of possible solutions for the trigonometric equation sin²x + sinx - 6 = 0. To find the solutions, we can treat this as a quadratic equation by substituting u for sinx, turning it into u² + u - 6 = 0. We can then factor this equation or use the quadratic formula to find the values of u, and subsequently, the values of x. Remember that since sin(x) can only have values between -1 and +1, we need to check which of the solutions fall within this range before determining the number of solutions in the original interval considered for x (usually between 0 and 2π for a single period of the sine function).