1.05o Trigonometric equations: solve in given intervals

1022 questions

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Edexcel C2 2007 January Q6
6 marks Standard +0.3
6. Find all the solutions, in the interval \(0 \leqslant x < 2 \pi\), of the equation $$2 \cos ^ { 2 } x + 1 = 5 \sin x$$ giving each solution in terms of \(\pi\).
Edexcel C2 2009 January Q8
8 marks Moderate -0.3
8.
  1. Show that the equation $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ can be written as $$4 \cos ^ { 2 } x - 9 \cos x + 2 = 0$$
  2. Hence solve, for \(0 \leqslant x < 720 ^ { \circ }\), $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ giving your answers to 1 decimal place.
Edexcel C2 2010 January Q2
6 marks Moderate -0.3
2.
  1. Show that the equation $$5 \sin x = 1 + 2 \cos ^ { 2 } x$$ can be written in the form $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$
  2. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$
Edexcel C2 2011 January Q7
7 marks Moderate -0.3
  1. Show that the equation $$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ can be written in the form $$4 \sin ^ { 2 } x + 7 \sin x + 3 = 0$$
  2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2012 January Q9
10 marks Standard +0.3
  1. Find the solutions of the equation \(\sin \left( 3 x - 15 ^ { \circ } \right) = \frac { 1 } { 2 }\), for which \(0 \leqslant x \leqslant 180 ^ { \circ }\) (6)
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42116a65-60ec-4dff-a05e-bab529939e1e-13_476_1141_495_406} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows part of the curve with equation $$y = \sin ( a x - b ) , \text { where } a > 0,0 < b < \pi$$ The curve cuts the \(x\)-axis at the points \(P , Q\) and \(R\) as shown.
    Given that the coordinates of \(P , Q\) and \(R\) are \(\left( \frac { \pi } { 10 } , 0 \right) , \left( \frac { 3 \pi } { 5 } , 0 \right)\) and \(\left( \frac { 11 \pi } { 10 } , 0 \right)\) respectively, find the values of \(a\) and \(b\).
Edexcel C2 2013 January Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leqslant x < 180 ^ { \circ }\), $$\cos \left( 3 x - 10 ^ { \circ } \right) = - 0.4$$ giving your answers to 1 decimal place. You should show each step in your working.
Edexcel C2 2014 January Q5
7 marks Moderate -0.8
5. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.
  1. Show that the height of the water 1 hour after midnight is 12.5 metres.
  2. Find, to the nearest minute, the times before midday when the height of the water is 9 metres.
Edexcel C2 2014 January Q9
9 marks Standard +0.3
9.
  1. Show that the equation $$5 \sin x - \cos ^ { 2 } x + 2 \sin ^ { 2 } x = 1$$ can be written in the form $$3 \sin ^ { 2 } x + 5 \sin x - 2 = 0$$
  2. Hence solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), the equation $$5 \sin 2 \theta - \cos ^ { 2 } 2 \theta + 2 \sin ^ { 2 } 2 \theta = 1$$ giving your answers to 2 decimal places.
Edexcel C2 2005 June Q5
8 marks Moderate -0.8
5. Solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), the equation
  1. \(\quad \sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
  2. \(\cos 2 x = - 0.9\), giving your answers to 1 decimal place.
Edexcel C2 2006 June Q6
4 marks Moderate -0.8
6.
  1. Given that \(\sin \theta = 5 \cos \theta\), find the value of \(\tan \theta\).
  2. Hence, or otherwise, find the values of \(\theta\) in the interval \(0 \leqslant \theta < 360 ^ { \circ }\) for which $$\sin \theta = 5 \cos \theta ,$$ giving your answers to 1 decimal place.
Edexcel C2 2007 June Q9
10 marks Moderate -0.8
9.
  1. Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
  2. Write down the exact coordinates of the points where the graph meets the coordinate axes.
  3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$ giving your answers in radians to 2 decimal places.
Edexcel C2 2008 June Q9
10 marks Moderate -0.8
9. Solve, for \(0 \leqslant x < 360 ^ { \circ }\),
  1. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\)
  2. \(\cos 3 x = - \frac { 1 } { 2 }\)
Edexcel C2 2009 June Q7
10 marks Moderate -0.3
7.
  1. Solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0$$
  2. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \sin x = 3 \tan x .$$
Edexcel C2 2010 June Q5
6 marks Moderate -0.3
5.
  1. Given that \(5 \sin \theta = 2 \cos \theta\), find the value of \(\tan \theta\).
  2. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin 2 x = 2 \cos 2 x$$ giving your answers to 1 decimal place.
Edexcel C2 2011 June Q7
10 marks Moderate -0.3
  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.
    You must show clearly how you obtained your answers.
Edexcel C2 2012 June Q6
7 marks Moderate -0.3
  1. Show that the equation $$\tan 2 x = 5 \sin 2 x$$ can be written in the form $$( 1 - 5 \cos 2 x ) \sin 2 x = 0$$
  2. Hence solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), $$\tan 2 x = 5 \sin 2 x$$ giving your answers to 1 decimal place where appropriate.
    You must show clearly how you obtained your answers.
Edexcel C2 2013 June Q9
12 marks Standard +0.3
  1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$ giving your answers to 1 decimal place.
  2. Find all the values of \(x\), in the interval \(0 \leqslant x < 360 ^ { \circ }\), for which $$9 \cos ^ { 2 } x - 11 \cos x + 3 \sin ^ { 2 } x = 0$$ giving your answers to 1 decimal place. You must show clearly how you obtained your answers.
Edexcel C2 2013 June Q8
11 marks Standard +0.3
8.
  1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
  2. (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.
Edexcel C2 2014 June Q7
8 marks Standard +0.3
7.
  1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$\frac { \sin 2 \theta } { ( 4 \sin 2 \theta - 1 ) } = 1$$ giving your answers to 1 decimal place.
  2. Solve, for \(0 \leqslant x < 2 \pi\), the equation $$5 \sin ^ { 2 } x - 2 \cos x - 5 = 0$$ giving your answers to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2014 June Q7
9 marks Moderate -0.3
7.
  1. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \sin \left( \theta + 60 ^ { \circ } \right) = 4$$ giving your answers to 1 decimal place.
    You must show each step of your working.
  2. Solve, for \(- \pi \leqslant x < \pi\), the equation $$2 \tan x - 3 \sin x = 0$$ giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C2 2015 June Q8
9 marks Moderate -0.3
8.
  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 \(0 \leqslant x < 360 ^ { \circ }\)
Edexcel C2 2016 June Q6
9 marks Moderate -0.3
6.
  1. Solve, for \(- \pi < \theta \leqslant \pi\), $$1 - 2 \cos \left( \theta - \frac { \pi } { 5 } \right) = 0$$ giving your answers in terms of \(\pi\).
  2. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos ^ { 2 } x + 7 \sin x - 2 = 0$$ giving your answers to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C2 2017 June Q8
8 marks Moderate -0.3
8.
  1. Show that the equation $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ can be written in the form $$( 3 \sin x - 1 ) ^ { 2 } = 2$$
  2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ giving your answers to 2 decimal places.
Edexcel C2 2018 June Q8
9 marks Moderate -0.3
8 In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos \left( x + 70 ^ { \circ } \right) = 3$$ giving your answers in degrees to one decimal place.
  2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$6 \cos ^ { 2 } \theta - 5 = 6 \sin ^ { 2 } \theta + \sin \theta$$ giving your answers in radians to 3 significant figures.
Edexcel C2 Specimen Q4
7 marks Standard +0.3
4. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(3 \sin ^ { 2 } x = 1 + \cos x\), giving your answers to the nearest degree.