10. A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)