Final answer:
The solution to the trigonometric inequality 2 - 3csc(x) > 8 over the interval [0, π/6] is a. 0 < x < π/6.
Step-by-step explanation:
The given inequality is 2 - 3csc(x) > 8. We can solve this inequality by first rearranging it to get -3csc(x) > 6. Next, we divide both sides of the inequality by -3, but we need to flip the inequality sign since we are dividing by a negative number. This gives us csc(x) < -2.
Now, we know that csc(x) is equal to 1/sin(x). So, we have sin(x) > -1/2. We can find the interval in which this inequality is true by looking at the unit circle and finding the values of x where sin(x) is greater than -1/2. From the unit circle, we can see that the values of x that satisfy this inequality are in the interval (-π/6 , π/6).
Therefore, the solution to the trigonometric inequality 2 - 3csc(x) > 8 over the interval [0, π/6] is a. 0 < x < π/6.