6 A curve has parametric equations
$$x = \ln ( t + 1 ) , \quad y = t ^ { 2 } \ln t$$
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
- Find the exact value of \(t\) at the stationary point.
- Find the gradient of the curve at the point where it crosses the \(x\)-axis.
- Express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers.
- Hence express \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta )\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
- Using the result of part (ii), solve the equation \(\sin 2 \theta ( 3 \sec \theta + 4 \operatorname { cosec } \theta ) = 7\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).