1.05l Double angle formulae: and compound angle formulae

575 questions

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CAIE P3 2022 June Q3
6 marks Standard +0.8
3 Solve the equation \(2 \cot 2 x + 3 \cot x = 5\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
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 2022 June Q2
5 marks Moderate -0.3
2 Solve the equation \(\cos \left( \theta - 60 ^ { \circ } \right) = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2023 June Q4
6 marks Moderate -0.3
4
  1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
  2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2023 June Q4
5 marks Standard +0.8
4 Solve the equation \(2 \cos x - \cos \frac { 1 } { 2 } x = 1\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2024 June Q7
9 marks Standard +0.3
7
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta \equiv \cos 2 \theta\).
  2. Hence find the exact value of \(\int _ { - \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta + 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \right) \mathrm { d } \theta\).
CAIE P3 2020 March Q6
8 marks Standard +0.3
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + 4 y ^ { 2 } } { \mathrm { e } ^ { x } }$$ It is given that \(y = 0\) when \(x = 1\).
  1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
  2. State what happens to the value of \(y\) as \(x\) tends to infinity.
CAIE P3 2022 March Q5
6 marks Standard +0.8
5 The angles \(\alpha\) and \(\beta\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) and are such that $$\tan ( \alpha + \beta ) = 2 \quad \text { and } \quad \tan \alpha = 3 \tan \beta .$$ Find the possible values of \(\alpha\) and \(\beta\).
CAIE P3 2020 November Q4
6 marks Standard +0.3
4
  1. Show that the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\) can be written in the form $$\tan ^ { 2 } \theta + 3 \sqrt { 3 } \tan \theta - 2 = 0$$
  2. Hence solve the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2021 November Q5
6 marks Standard +0.3
5
  1. Show that the equation $$\cot 2 \theta + \cot \theta = 2$$ can be expressed as a quadratic equation in \(\tan \theta\).
  2. Hence solve the equation \(\cot 2 \theta + \cot \theta = 2\), for \(0 < \theta < \pi\), giving your answers correct to 3 decimal places.
CAIE P3 2021 November Q6
6 marks Standard +0.3
6
  1. Using the expansions of \(\sin ( 3 x + 2 x )\) and \(\sin ( 3 x - 2 x )\), show that $$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )\).
CAIE P3 2021 November Q8
7 marks Standard +0.3
8
  1. By first expanding \(\left( \cos ^ { 2 } \theta + \sin ^ { 2 } \theta \right) ^ { 2 }\), show that $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta \equiv 1 - \frac { 1 } { 2 } \sin ^ { 2 } 2 \theta .$$
  2. Hence solve the equation $$\cos ^ { 4 } \theta + \sin ^ { 4 } \theta = \frac { 5 } { 9 } ,$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2021 November Q5
5 marks Standard +0.3
5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2021 November Q6
7 marks Standard +0.3
6
  1. By first expanding \(\cos \left( x - 60 ^ { \circ } \right)\), show that the expression $$2 \cos \left( x - 60 ^ { \circ } \right) + \cos x$$ can be written in the form \(R \cos ( x - \alpha )\), 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.
  2. Hence find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which \(2 \cos \left( x - 60 ^ { \circ } \right) + \cos x\) takes its least possible value.
CAIE P3 2022 November Q4
5 marks Standard +0.8
4 Solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 2 \cot x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2022 November Q6
7 marks Standard +0.8
6
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta + 3 \equiv 8 \cos ^ { 4 } \theta\).
  2. Hence solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2023 November Q5
6 marks Standard +0.3
5
  1. Given that $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ find the exact value of \(\tan x\).
  2. Hence find the exact roots of the equation $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P3 2023 November Q7
8 marks Standard +0.3
7
  1. By expressing \(3 \theta\) as \(2 \theta + \theta\), prove the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Hence solve the equation $$\cos 3 \theta + \cos \theta \cos 2 \theta = \cos ^ { 2 } \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2023 November Q6
6 marks Standard +0.8
6
  1. Show that the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\) can be written in the form $$4 \sin ^ { 4 } \theta + 3 \sin ^ { 2 } \theta - 1 = 0$$
  2. Hence solve the equation \(\cot ^ { 2 } \theta + 2 \cos 2 \theta = 4\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2024 November Q5
6 marks Standard +0.3
5
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
  2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).
CAIE Further Paper 2 2022 June Q8
16 marks Challenging +1.8
8
  1. Find \(\int \sin \theta \cos ^ { n } \theta d \theta\), where \(n \neq - 1\).
    Let \(I _ { m , n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { m } \theta \cos ^ { n } \theta d \theta\).
  2. Show that, for \(m \geqslant 2\) and \(n \geqslant 0\), $$I _ { m , n } = \frac { m - 1 } { m + n } I _ { m - 2 , n }$$
  3. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 5 } \theta = a \cos 5 \theta + b \cos 3 \theta + c \cos \theta$$ where \(a\), \(b\) and \(c\) are constants to be determined.
  4. Using the results given in parts (b) and (c), find the exact value of \(I _ { 2,5 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2024 November Q8
14 marks Hard +2.3
8
  1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
    Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
  2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-18_2716_40_109_2009}
  3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2020 Specimen Q6
10 marks Challenging +1.8
6
  1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
Edexcel P2 2019 June Q9
8 marks Standard +0.3
9.
  1. Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written in the form $$5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0$$
  2. Hence solve, for \(0 \leqslant x < \frac { \pi } { 2 }\) $$\cos 2 x - 1 = 4 \sin 2 x \tan 2 x$$ giving your answers, where appropriate, to 2 decimal places.
Edexcel P2 2021 June Q8
10 marks Standard +0.3
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
  2. (a) Show that the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ can be written in the form $$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$ where \(a , b\) and \(c\) are constants to be found.
    (b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-27_2644_1840_118_111}