Given an isosceles triangle with side lengths $a$, $a$, and $b$, with the vertex angle $\theta$ opposite the side $b$. The sides and angle are related by the determinant of the following matrix being zero:
\[
\begin{vmatrix}
\sin \theta & \cos \theta & 1 \\
\cos \theta & \sin \theta & 1 \\
a & b & 1
\end{vmatrix}
= 0.
\]
Find the angle $\theta$ in degrees. Let's think step by step and output the final answer within \boxed{}.
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