8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e14881c1-5ba5-4868-92ee-8bc58d4884dc-13_808_965_248_502}
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\caption{Figure 3}
\end{figure}
The curve shown in Figure 3 has parametric equations
$$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$
The point \(A\), with coordinates ( \(k , 1\) ), lies on the curve.
Given that \(k > 0\)
- find the exact value of \(k\),
- find the gradient of the curve at the point \(A\).
There is one point on the curve where the gradient is equal to \(- \frac { 1 } { 2 }\)
- Find the value of \(t\) at this point, showing each step in your working and giving your answer to 4 decimal places.
[0pt]
[Solutions based entirely on graphical or numerical methods are not acceptable.]