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The diagram shows the curve with equation \(x = \left( 37 + 10 y - 2 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\).
- Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
- Hence find the equation of the tangent to the curve at the point ( 7,3 ), giving your answer in the form \(y = m x + c\).
- Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
- Hence
(a) solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(8 \sin \theta - 6 \cos \theta = 9\),
(b) find the greatest possible value of
$$32 \sin x - 24 \cos x - ( 16 \sin y - 12 \cos y )$$
as the angles \(x\) and \(y\) vary.