Solve equation directly given harmonic form

A question is this type if and only if the harmonic form R·sin(θ ± α) or R·cos(θ ± α) is already given or stated, and the task is only to solve an equation using it.

2 questions · Standard +0.0

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OCR C3 2010 June Q8
9 marks Standard +0.3
  1. Express \(3 \cos x + 3 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [3]
  2. The expression T\((x)\) is defined by T\((x) = \frac{8}{3 \cos x + 3 \sin x}\).
    1. Determine a value of \(x\) for which T\((x)\) is not defined. [2]
    2. Find the smallest positive value of \(x\) satisfying T\((3x) = \frac{8}{3}\sqrt{6}\), giving your answer in an exact form. [4]
SPS SPS SM Pure 2023 October Q4
12 marks Moderate -0.3
$$f(x) = 12 \cos x - 4 \sin x.$$ Given that \(f(x) = R \cos(x + \alpha)\), where \(R \geq 0\) and \(0 \leq \alpha \leq 90°\),
  1. find the value of \(R\) and the value of \(\alpha\). [4]
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leq x < 360°\), giving your answers to one decimal place. [5]
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\). [1]
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. [2]