1. The problem asks us to determine whether given trigonometric statements are possible or impossible. 2. We are given an example: $\sin \theta = \frac{1}{2}$ and $\csc \theta = 2$. 3. Recall the definition: $\csc \theta = \frac{1}{\sin \theta}$. 4. Using this, if $\sin \theta = \frac{1}{2}$, then $\csc \theta = \frac{1}{\frac{1}{2}} = 2$. 5. Since the given $\csc \theta = 2$ matches the calculated value, this statement is possible. 6. To determine if other statements are possible, check if the reciprocal relationship between sine and cosecant holds true. 7. Also, remember sine values must be between $-1$ and $1$, so any sine value outside this range is impossible. 8. Similarly, cosecant values must be $\leq -1$ or $\geq 1$ because it is the reciprocal of sine. 9. In summary, verify if $\csc \theta = \frac{1}{\sin \theta}$ and if sine values are within $[-1,1]$ to decide possibility.