We apply Newton's second law to each mass. Let the downward direction for $m_2$ and the up-the-incline direction for $m_1$ be positive. The acceleration of both blocks has magnitude $a$.

For the hanging mass $m_2$: The net force is the sum of gravity ($m_2 g$), the external force ($F$), and tension ($T$).

$$m_2 g - T - F = m_2 a$$

Solving for the tension $T$:

$$T = m_2 g - F - m_2 a = m_2(g - a) - F$$

For the mass on the incline $m_1$: The net force along the incline is the sum of tension ($T$) and the component of gravity parallel to the incline ($m_1 g \sin\beta$).

$$T - m_1 g \sin\beta = m_1 a$$

Solving for $\sin\beta$:

$$\sin\beta = \frac{T - m_1 a}{m_1 g}$$

Substituting the expression for $T$ from the first equation:

$$\sin\beta = \frac{(m_2(g - a) - F) - m_1 a}{m_1 g}$$

The angle $\beta$ is the arcsin of this expression. For a physical solution, the argument of the arcsin must be between 0 and 1.