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The problem provided asks to find the exact values of certain trigonometric expressions under given conditions.

You are asked to find:

- \( \sin(\alpha + \beta) \)
- Given:
  - \( \frac{7}{\sin\left(\frac{\pi}{7}\right)} \)
  - \( 0 < \alpha < \frac{\pi}{2} \)
  - \( \cos\left(\frac{61^\circ}{180^\circ} \pi \right) \)
  - \( \tan\left(\alpha - \beta\right) \)
  - \(\alpha > 0\), \(\beta < 0\).

Since the exact details of \( \alpha \) and \( \beta \) seem to relate to sum and difference identities, we would use the trigonometric sum formulas:

- **Sine sum identity**: 
  \[
  \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)
  \]
  
Could you clarify the values of \( \alpha \) and \( \beta \) based on this? If you'd like me to assist with any specific identity, feel free to ask!

Do you need help solving this step-by-step?

Here are some relative questions:
1. How can we calculate \( \sin(\alpha) \) and \( \cos(\alpha) \) using known values?
2. Do you need help applying the cosine rule for the given angle \( \beta \)?
3. What is the simplified form of \( \sin(\alpha + \beta) \) for specific angle measures?
4. Would you like more details on trigonometric identities used here?
5. How do angle values affect the sum and difference formulas in trigonometry?

**Tip**: Always double-check angle units (degrees vs. radians) when applying formulas!

Find the exact value of each of the following under the given conditions: (a) sin(α + β), (b) cos(α + β), (c) sin(α − β), (d) tan(α − β).

Solve trigonometric expressions involving sum and difference formulas for sine, cosine, and tangent under specific conditions. This problem involves key trigonometric identities and is suitable for high school students in Grades 10-12.

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