4 The parametric equations of a curve are
$$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$
where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
- Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
- Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 .
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The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.