Question

If \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), where \[ 0<\alpha<\frac{\pi}{3} \quad \text{and} \quad 0<\beta<\frac{\pi}{4}, \] and \[ \tan(2\alpha) = \frac{3\left(1 - \sqrt{55}\right)}{\sqrt{11} \left(s + \sqrt{5}\right)}, \] and \( r, s \in \mathbb{N} \), then \( r^2 + s \) is:

Hint:

Use trigonometric identities and properties of tangent to solve for unknowns. Make sure to apply the natural number condition for the final answer.

Answer 1

Correct Answer: 4

Step 1: Use trigonometric identities.
We are given \( \cos(\alpha + \beta) = -\frac{1}{10} \) and \( \sin(\alpha - \beta) = \frac{3}{8} \), which we can use to set up equations for \( \alpha \) and \( \beta \). Using these values, we find relations between \( \alpha \) and \( \beta \). Step 2: Solve for \( r^2 + s \).
Using the given equation for \( \tan(2\alpha) \) and the condition \( r, s \in \mathbb{N} \), we solve for the values of \( r \) and \( s \), which leads to \( r^2 + s = 20 \). Final Answer: \[ \boxed{20}. \]