Question 9 - A-Level Maths

Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
Hence
(a) solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2 \pi\),
(b) find \(\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).