1.05o Trigonometric equations: solve in given intervals

1022 questions

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Edexcel C12 2019 June Q12
8 marks Standard +0.3
12.
  1. Show that $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ may be expressed in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ where \(a , b\) and \(c\) are constants to be found.
  2. Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ giving your answers in radians to 2 decimal places.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}
Edexcel C12 2016 October Q10
8 marks Standard +0.3
10.
  1. Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$
  2. Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}
Edexcel C12 2017 October Q7
9 marks Standard +0.3
7. $$g ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 18 x - 8$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\),
  1. show that \(a = - 3\)
  2. Hence, using algebra, fully factorise \(\mathrm { g } ( x )\). Using your answer to part (b),
  3. solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$2 \sin ^ { 3 } \theta - 3 \sin ^ { 2 } \theta - 18 \sin \theta = 8$$ giving each answer, in radians, as a multiple of \(\pi\).
Edexcel C12 2017 October Q12
11 marks Standard +0.3
12.
  1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
  2. (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.
Edexcel C12 2018 October Q12
8 marks Standard +0.3
12.
  1. Show that the equation $$6 \cos x - 5 \tan x = 0$$ may be expressed in the form $$6 \sin ^ { 2 } x + 5 \sin x - 6 = 0$$
  2. Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$6 \cos \left( 2 \theta - 10 ^ { \circ } \right) - 5 \tan \left( 2 \theta - 10 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 Specimen Q14
10 marks Moderate -0.3
In this question you must show all stages of your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place, $$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$
  2. Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-35_108_95_2572_1804}
Edexcel P2 2020 January Q3
8 marks Standard +0.3
3. $$f ( x ) = 6 x ^ { 3 } + 17 x ^ { 2 } + 4 x - 12$$
  1. Use the factor theorem to show that ( \(2 x + 3\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Hence, using algebra, write \(\mathrm { f } ( x )\) as a product of three linear factors.
  3. Solve, for \(\frac { \pi } { 2 } < \theta < \pi\), the equation $$6 \tan ^ { 3 } \theta + 17 \tan ^ { 2 } \theta + 4 \tan \theta - 12 = 0$$ giving your answers to 3 significant figures.
Edexcel P2 2020 January Q7
7 marks Standard +0.3
7.
  1. Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$
  2. Hence solve, for \(0 \leqslant x \leqslant 90 ^ { \circ }\), the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.
Edexcel P2 2021 January Q6
8 marks Standard +0.3
  1. Show that the equation $$\frac { 3 \sin \theta \cos \theta } { 2 \sin \theta - 1 } = 5 \tan \theta \quad \sin \theta \neq \frac { 1 } { 2 }$$ can be written in the form $$3 \sin ^ { 3 } \theta + 10 \sin ^ { 2 } \theta - 8 \sin \theta = 0$$
  2. Hence solve, for \(- \frac { \pi } { 4 } < x < \frac { \pi } { 4 }\) $$\frac { 3 \sin 2 x \cos 2 x } { 2 \sin 2 x - 1 } = 5 \tan 2 x$$ giving your answers to 3 decimal places where appropriate.
Edexcel P2 2022 January Q7
8 marks Standard +0.3
7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation $$3 \sin \left( 2 x - 15 ^ { \circ } \right) = \cos \left( 2 x - 15 ^ { \circ } \right)$$ giving your answers to one decimal place.
  2. Solve, for \(0 < \theta < 2 \pi\), the equation $$4 \sin ^ { 2 } \theta + 8 \cos \theta = 3$$ giving your answers to 3 significant figures.
Edexcel P2 2023 January Q8
9 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable.
  1. Solve, for \(- \frac { \pi } { 2 } < x < \pi\), the equation $$5 \sin ( 3 x + 0.1 ) + 2 = 0$$ giving your answers, in radians, to 2 decimal places.
  2. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$2 \tan \theta \sin \theta = 5 + \cos \theta$$ giving your answers, in degrees, to one decimal place.
Edexcel P2 2024 January Q9
14 marks Standard +0.3
  1. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
    The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
    Given that the \(y\) coordinate of \(P\) is 7
  1. state the value of \(A\),
  2. find the exact coordinates of \(Q\),
  3. find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.
Edexcel P2 2019 June Q6
8 marks Standard +0.3
6. \(\mathrm { f } ( x ) = k x ^ { 3 } - 15 x ^ { 2 } - 32 x - 12\) where \(k\) is a constant Given ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(k = 9\)
  2. Using algebra and showing each step of your working, fully factorise \(\mathrm { f } ( x )\).
  3. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \cos ^ { 3 } \theta - 15 \cos ^ { 2 } \theta - 32 \cos \theta - 12 = 0$$ giving your answers to one decimal place.
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}
Edexcel P2 2023 June Q9
7 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
  2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.
Edexcel P2 2024 June Q8
12 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < x \leqslant \pi\), the equation $$5 \sin x \tan x + 13 = \cos x$$ giving your answer in radians to 3 significant figures.
  2. The temperature inside a greenhouse is monitored on one particular day. The temperature, \(H ^ { \circ } \mathrm { C }\), inside the greenhouse, \(t\) hours after midnight, is modelled by the equation $$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(k\) is a constant.
    Use the equation of the model to answer parts (a) to (c).
    Given that
    • the temperature inside the greenhouse was \(20 ^ { \circ } \mathrm { C }\) at 6 am
    • \(0 < k < 20\) (a) find all possible values for \(k\), giving each answer to 2 decimal places.
    Given further that \(0 < k < 10\) (b) find the maximum temperature inside the greenhouse,
    (c) find the time of day at which this maximum temperature occurs. Give your answer to the nearest minute.
Edexcel P2 2019 October Q9
12 marks Standard +0.3
9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.
  1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$ giving your answers to one decimal place.
  2. The first three terms of an arithmetic sequence are $$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$ where \(\alpha\) is a constant.
    1. Show that \(2 \cos \alpha = 3 \sin ^ { 2 } \alpha\) Given that \(\pi < \alpha < 2 \pi\),
    2. find, showing all working, the value of \(\alpha\) to 3 decimal places.
Edexcel P2 2020 October Q7
7 marks Standard +0.3
7.
  1. Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$
  2. Solve, for \(0 \leqslant x < 90 ^ { \circ }\), the equation $$3 \cos ^ { 2 } \left( 2 x + 10 ^ { \circ } \right) = 1$$ giving your answers in degrees to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel P2 2021 October Q10
10 marks Standard +0.3
10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) $$\tan ^ { 2 } \left( 2 x + \frac { \pi } { 4 } \right) = 3$$
  2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$( 2 \sin \theta - \cos \theta ) ^ { 2 } = 1$$ giving your answers, as appropriate, to one decimal place.
Edexcel P2 2022 October Q5
8 marks Standard +0.3
  1. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
  2. Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$
Edexcel P2 2023 October Q3
7 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) the equation $$2 \tan \theta + 3 \sin \theta = 0$$ giving your answers, as appropriate, to one decimal place.
  2. Hence, or otherwise, find the smallest positive solution of $$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$ giving your answer to one decimal place.
Edexcel P2 2018 Specimen Q9
9 marks Moderate -0.3
9.
  1. Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\)
  2. Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$
    1. find \(\cos x\) in terms of \(k\)
    2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-30_2671_1942_107_121}
Edexcel C2 2005 January Q4
7 marks Moderate -0.3
4.
  1. Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$
  2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2006 January Q8
9 marks Moderate -0.8
  1. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$
  2. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$