5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-16_938_1257_125_486}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with parametric equations
$$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$
The point \(A\) lies on the curve \(C\).
Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)
- find the exact value of \(k\), giving your answer in a fully simplified form.
- Find the equation of the tangent to \(C\) at the point \(A\).
Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.