Challenging +1.2 This is an implicit differentiation problem requiring students to find where dy/dx = 0. While it involves implicit differentiation of trigonometric functions and solving sin x = 0 within given bounds, the steps are methodical: differentiate implicitly, set dy/dx = 0, solve for x values, then find corresponding y values from the original equation. The trigonometric equation solving is straightforward (sin x = 0 gives x = 0, π) and finding y from cos y = 0.5 is standard. This is above-average difficulty due to the implicit differentiation and multi-step nature, but remains a recognizable Further Maths technique without requiring novel insight.
A curve \(C\) is given by the equation
$$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$
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.)
[5]
A curve $C$ is given by the equation
$$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$
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.)
[5]
\hfill \mbox{\textit{SPS SPS FM 2020 Q10 [5]}}