7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-12_556_860_246_452}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows the curve with parametric equations
$$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$
The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).
- Find the value of the parameter \(t\) at \(P\).
- Show that the tangent to the curve at \(P\) has the equation
$$y = 2 x + 1$$
The shaded region is bounded by the curve, the coordinate axes and the line \(x = \frac { 1 } { 2 }\).
- Show that the area of the shaded region is given by
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$
where \(k\) is a positive integer to be found.
- Hence find the exact area of the shaded region.
7. continued
7. continued