Range of squared harmonic expression

A question is this type if and only if it asks for the maximum, minimum, or range of values of an expression involving the square of a harmonic form, such as (a·sin + b·cos)² or expressions like c + d·(harmonic)².

1 questions · Moderate -0.3

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CAIE P3 2021 November Q2
5 marks Moderate -0.3
2
  1. Express \(5 \sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence state the greatest and least possible values of \(( 5 \sin x - 3 \cos x ) ^ { 2 }\).