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!