- (a) Express \(2 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
The first three terms of an arithmetic sequence are
$$\cos x \quad \cos x + \sin x \quad \cos x + 2 \sin x \quad x \neq n \pi$$
Given that \(S _ { 9 }\) represents the sum of the first 9 terms of this sequence as \(x\) varies,
(b) (i) find the exact maximum value of \(S _ { 9 }\)
(ii) deduce the smallest positive value of \(x\) at which this maximum value of \(S _ { 9 }\) occurs.