Expand then express in harmonic form

A question is this type if and only if it explicitly requires first expanding a compound angle expression (e.g., sin(x-30°)) before expressing the result in harmonic form.

3 questions · Standard +0.3

1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc
Sort by: Default | Easiest first | Hardest first
CAIE P3 2017 June Q8
8 marks Standard +0.3
8
  1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR C3 2008 June Q8
10 marks Standard +0.3
8 The expression \(\mathrm { T } ( \theta )\) is defined for \(\theta\) in degrees by $$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$
  1. Express \(\mathrm { T } ( \theta )\) in the form \(A \sin \theta + B \cos \theta\), giving the exact values of the constants \(A\) and \(B\). [3]
  2. Hence express \(\mathrm { T } ( \theta )\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  3. Find the smallest positive value of \(\theta\) such that \(\mathrm { T } ( \theta ) + 1 = 0\).
CAIE P3 2013 June Q7
9 marks Standard +0.3
  1. By first expanding \(\cos(x + 45°)\), express \(\cos(x + 45°) - (\sqrt{2}) \sin x\) in the form \(R \cos(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$\cos(x + 45°) - (\sqrt{2}) \sin x = 2,$$ for \(0° < x < 360°\). [4]