- (a) Express \(35 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\),
$$70 \sin x - 24 \cos x = 37$$
(Solutions based entirely on graphical or numerical methods are not acceptable.)
$$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$
(c) Use your answer to part (a) to calculate
- the minimum value of \(y\),
- the smallest value of \(x , x > 0\), at which this minimum value occurs.