Double angle equations requiring identity expansion and factorisation

Equations mixing double angle terms with single angle terms (e.g. sin θ = 2cos2θ + 1, cosθ + 4cos2θ = 3, 3sin2x = cosx) requiring expansion of the double angle identity and algebraic manipulation such as factorisation or forming a quadratic.

17 questions · Standard +0.1

1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals
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CAIE P2 2016 June Q2
5 marks Standard +0.3
2 Solve the equation \(5 \tan 2 \theta = 4 \cot \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2010 June Q2
6 marks Standard +0.3
2 Solve the equation $$\sin \theta = 2 \cos 2 \theta + 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2011 June Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \theta + 4 \cos 2 \theta = 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2003 November Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \theta + 3 \cos 2 \theta = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2004 November Q3
4 marks Moderate -0.3
3 Find the values of \(x\) satisfying the equation $$3 \sin 2 x = \cos x$$ for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P2 2012 November Q3
4 marks Moderate -0.3
3 Solve the equation $$2 \cos 2 \theta = 4 \cos \theta - 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
Edexcel C34 2015 January Q2
5 marks Standard +0.3
2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2011 January Q3
6 marks Moderate -0.3
  1. Find all the solutions of
$$2 \cos 2 \theta = 1 - 2 \sin \theta$$ in the interval \(0 \leqslant \theta < 360 ^ { \circ }\).
OCR MEI C4 2006 January Q4
6 marks Moderate -0.3
4 Solve the equation \(2 \sin 2 \theta + \cos 2 \theta = 1\), for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
OCR MEI C4 2008 June Q3
7 marks Moderate -0.3
3 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
Pre-U Pre-U 9794/1 2017 June Q4
4 marks Moderate -0.3
4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
OCR MEI C2 2008 June Q10
5 marks Standard +0.3
Showing your method, solve the equation \(2\sin^2\theta = \cos\theta + 2\) for values of \(\theta\) between \(0°\) and \(360°\). [5]
OCR C4 Q5
6 marks Standard +0.3
Solve the equation \(2\sin 2\theta + \cos 2\theta = 1\), for \(0° \leqslant \theta < 360°\). [6]
OCR C4 Q6
7 marks Standard +0.3
Express \(6\cos 2\theta + \sin\theta\) in terms of \(\sin\theta\). Hence solve the equation \(6\cos 2\theta + \sin\theta = 0\), for \(0° \leqslant \theta \leqslant 360°\). [7]
WJEC Unit 1 2024 June Q15
7 marks Standard +0.8
The diagram shows a sketch of part of the curve with equation \(y = 2\sin x + 3\cos^2 x - 3\). The curve crosses the \(x\)-axis at the points O, A, B and C. \includegraphics{figure_15} Find the value of \(x\) at each of the points A, B and C. [7]
OCR H240/03 2017 Specimen Q3
4 marks Standard +0.8
In this question you must show detailed reasoning. Given that \(5\sin 2x = 3\cos x\), where \(0° < x < 90°\), find the exact value of \(\sin x\). [4]
OCR AS Pure 2017 Specimen Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Solve the equation \(2\cos^2 x = 2 - \sin x\) for \(0° \leq x \leq 180°\). [5]