Solve equation with sin2x/cos2x by substitution

Solve a trigonometric equation containing cos2x or sin2x (but not tan2x) by substituting a double angle identity to reduce to a quadratic or simpler equation in a single trig function.

15 questions · Standard +0.2

1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals
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CAIE P2 2017 November Q2
5 marks Standard +0.3
2 Solve the equation \(5 \cos \theta ( 1 + \cos 2 \theta ) = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2022 June Q2
5 marks Moderate -0.3
2 Solve the equation \(3 \cos 2 \theta = 3 \cos \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2021 November Q5
5 marks Standard +0.3
5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
Edexcel P3 2020 October Q1
5 marks Moderate -0.3
  1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation
$$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2009 June Q8
6 marks Standard +0.3
8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
(b) Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.
OCR C3 Q5
8 marks Standard +0.3
  1. (i) Find the exact value of \(x\) such that
$$3 \tan ^ { - 1 } ( x - 2 ) + \pi = 0$$ (ii) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).
OCR MEI C4 Q1
4 marks Moderate -0.3
1 Solve the equation \(2 \sin 2 \theta = \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR C4 Q4
7 marks Moderate -0.3
4 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).
OCR MEI C4 Q6
7 marks Moderate -0.3
6 Solve the equation \(2 \cos 2 x = 1 + \cos x\), for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
Edexcel AEA 2002 Specimen Q3
12 marks Challenging +1.8
3.Solve for values of \(\theta\) ,in degrees,in the range \(0 \leq \theta \leq 360\) , $$\sqrt { } 2 \times ( \sin 2 \theta + \cos \theta ) + \cos 3 \theta = \sin 2 \theta + \cos \theta$$
Edexcel AEA 2005 June Q2
8 marks Challenging +1.2
2.Solve,for \(0 < \theta < 2 \pi\) , $$\sin 2 \theta + \cos 2 \theta + 1 = \sqrt { 6 } \cos \theta$$ giving your answers in terms of \(\pi\) .
Edexcel AEA 2007 June Q3
11 marks Standard +0.8
3.(a)Solve,for \(0 \leq x < 2 \pi\) , $$\cos x + \cos 2 x = 0$$ (b)Find the exact value of \(x , x \geq 0\) ,for which $$\arccos x + \arccos 2 x = \frac { \pi } { 2 }$$ [ \(\arccos x\) is an alternative notation for \(\cos ^ { - 1 } x\) .]
OCR MEI C4 2015 June Q2
7 marks Moderate -0.3
2 Express \(6 \cos 2 \theta + \sin \theta\) in terms of \(\sin \theta\).
Hence solve the equation \(6 \cos 2 \theta + \sin \theta = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C4 2005 June Q5
7 marks Moderate -0.3
5 Solve the equation \(2 \cos 2 x = 1 + \cos x\), for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
AQA C4 2009 January Q5
9 marks Standard +0.3
5
  1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
  2. Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) to the nearest \(0.1 ^ { \circ }\), where appropriate.
  3. Given that \(\sin 2 x + \cos 2 x = 1 + \sin x\) and \(\sin x \neq 0\), show that \(2 ( \cos x - \sin x ) = 1\).