Final answer:
None of the provided options (a) θ = π/3, (b) θ = π/2, (c) θ = 2π/3, or (d) θ = π satisfies the equation csc²θ-9=0 with sinθ = 1/3 or sinθ = -1/3. As such, none of these options are correct for the given trigonometric equation.
Step-by-step explanation:
The question asks us to find all angles that satisfy the equation csc²θ-9=0. To solve this equation, we start by adding 9 to both sides to isolate the cosecant squared term, resulting in csc²θ = 9. Taking the square root of both sides, we get cscθ = ±3, which means sinθ = 1/3 or sinθ = -1/3 because cscθ is the reciprocal of sinθ. The sine function has a value of 1/3 at specific angles within the unit circle. However, for sinθ = -1/3, there are no standard angles in the unit circle where the sine has a value of -1/3.
Looking at the options provided:
- θ = π/3 (Option a) has a sine value of √3/2, which doesn't match 1/3.
- θ = π/2 (Option b) has a sine value of 1, not 1/3 or -1/3.
- θ = 2π/3 (Option c) has a sine value of √3/2, which also doesn't match 1/3.
- θ = π (Option d) has a sine value of 0, not 1/3 or -1/3.
Therefore, none of the mentioned options in the final answer are correct. The angle that satisfies csc²θ-9=0 with sinθ = 1/3 or sinθ = -1/3 is not represented by any of the provided options.