all solutions given x can be 0 all the way to pi 

Given the restriction that \(0 \leq x \leq \pi\), we will find all solutions within this interval from the general solutions:

\[
x = \frac{5\pi}{24} + n\pi \quad \text{or} \quad x = \frac{13\pi}{24} + n\pi, \quad \text{where } n \text{ is an integer.}
\]

### First Case: \(x = \frac{5\pi}{24} + n\pi\)
- For \(n = 0\): \(x = \frac{5\pi}{24} \)
- For \(n = 1\): \(x = \frac{5\pi}{24} + \pi = \frac{5\pi}{24} + \frac{24\pi}{24} = \frac{29\pi}{24} > \pi\) (not in the interval)

Therefore, only \(x = \frac{5\pi}{24}\) is a valid solution from this case.

### Second Case: \(x = \frac{13\pi}{24} + n\pi\)
- For \(n = 0\): \(x = \frac{13\pi}{24}\)
- For \(n = 1\): \(x = \frac{13\pi}{24} + \pi = \frac{13\pi}{24} + \frac{24\pi}{24} = \frac{37\pi}{24} > \pi\) (not in the interval)

Therefore, only \(x = \frac{13\pi}{24}\) is a valid solution from this case.

### Final Solutions:
The solutions in the interval \(0 \leq x \leq \pi\) are:
\[
x = \frac{5\pi}{24} \quad \text{and} \quad x = \frac{13\pi}{24}.
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you find the exact values of trigonometric functions at specific angles?
2. What are some methods to solve trigonometric equations in a restricted interval?
3. How does changing the interval affect the number of solutions in trigonometric equations?
4. What are the implications of the periodicity of trigonometric functions on solutions?
5. How can we graphically verify solutions to trigonometric equations?

### Tip:
When solving trigonometric equations with interval restrictions, always substitute back into the original equation to verify the validity of your solutions.

Explore how to find solutions to trigonometric equations within the interval 0 to pi. Learn about the periodic nature of trigonometric functions and how it affects the number of solutions. Includes detailed step-by-step solutions and related questions.

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