Question #41010

Let $A$ and $B$ be two events associated with a random experiment. The probability of event $A$ is given by $P(A) = \det(M)$, where $M$ is the $3 \times 3$ matrix: $M = \begin{pmatrix} \int_{-\pi/4}^{\pi/4} \ln\left(\frac{1-\sin x}{1+\sin x}\right) dx & \alpha^2+1 & \beta \\ -(\alpha^2+1) & \sum_{k=1}^{59} \left( \cos\frac{k\pi}{2} - \cos\frac{-k\pi}{2} \right) & \gamma \\ -\beta & -\gamma & \lim_{x \to 0} \frac{e^{x^2} - 1 - x^2}{x^4} - \frac{1}{2} \end{pmatrix}$ for some real constants $\alpha, \beta, \gamma$. Event $B$ is defined as the event that a number $t$ chosen at random from the sample space $S = \{t \in \mathbb{Z} : -59 \le t \le 283\}$ satisfies the inequality: $(t-59)^2 (t-283) < 0$ Find the value of $343 \times P(A^c \cap B)$.