The image contains mathematical inequalities and equations in Arabic script. Here is the transcription and solution process:

Transcription:

$\frac{1}{2} \leq \frac{2}{\sin(\theta)} \leq \frac{5}{3}$

To solve for $\theta$, we will handle the inequalities separately.

Steps:

1. Solve $\frac{1}{2} \leq \frac{2}{\sin(\theta)}$: $\frac{1}{2} \leq \frac{2}{\sin(\theta)}$ Multiplying both sides by $\sin(\theta)$ (assuming $\sin(\theta)$ is positive): $\frac{1}{2} \sin(\theta) \leq 2$ Rearranging: $\sin(\theta) \geq \frac{1}{4}$

2. Solve $\frac{2}{\sin(\theta)} \leq \frac{5}{3}$: $\frac{2}{\sin(\theta)} \leq \frac{5}{3}$ Multiplying both sides by $\sin(\theta)$ (assuming $\sin(\theta)$ is positive): $2 \leq \frac{5}{3} \sin(\theta)$ Rearranging: $\sin(\theta) \leq \frac{6}{5}$ Since $\sin(\theta) \leq 1$, this inequality always holds.

Combining the results from both parts: $\frac{1}{4} \leq \sin(\theta) \leq 1$

Solution:

$\theta = \sin^{-1}(x) \text{ for } \frac{1}{4} \leq x \leq 1$

Would you like further details or have any questions?

Here are some questions you might find helpful:

1. Can you explain the steps in solving trigonometric inequalities?
2. What are the properties of the sine function that are useful in solving these inequalities?
3. How do we handle cases where $\sin(\theta)$ might be negative?
4. Can you solve similar inequalities for other trigonometric functions?
5. How does the range of the sine function affect the solution of these inequalities?

Tip: Always check the domain and range of trigonometric functions when solving inequalities to ensure valid solutions.