Express in harmonic form

A question is this type if and only if it asks to express a linear combination of sin and cos (e.g., a·sin(θ) + b·cos(θ)) in the form R·sin(θ ± α) or R·cos(θ ± α), without requiring further solving.

4 questions · Moderate -0.4

1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

Sort by: Default | Easiest first | Hardest first

OCR MEI C4 2016 June Q1

6 marks Moderate -0.3

1 Express \(\cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence show that the equation \(\cos \theta - 3 \sin \theta = 4\) has no solution.

AQA Paper 2 2019 June Q6

6 marks Standard +0.8

A curve has equation $$y = a \sin x + b \cos x$$ where \(a\) and \(b\) are constants. The maximum value of \(y\) is 4 and the curve passes through the point \(\left(\frac{\pi}{3}, 2\sqrt{3}\right)\) as shown in the diagram. \includegraphics{figure_6} Find the exact values of \(a\) and \(b\). [6 marks]

AQA Paper 3 2020 June Q2

1 marks Easy -1.2

Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos (\theta + \alpha)$$ find the value of \(R\). Circle your answer. [1 mark] \(6\) \quad \(8\) \quad \(10\) \quad \(14\)

OCR MEI Paper 2 2022 June Q1

4 marks Moderate -0.8

Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R\) and \(\alpha\) are exact values to be determined. [4]