Standard +0.8 This requires implicit differentiation to find dy/dx, setting it to zero for horizontal tangents, then solving the resulting trigonometric system. While the techniques are standard Further Maths content, the question demands careful handling of multiple solutions across specified domains and justification of all cases, elevating it above routine exercises.
12. 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.) [0pt]
12. 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.)\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q12 [7]}}