1.05o Trigonometric equations: solve in given intervals

1022 questions

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OCR MEI Paper 3 Specimen Q8
8 marks Challenging +1.2
8 In Fig. 8, OAB is a thin bent rod, with \(\mathrm { OA } = 1 \mathrm {~m} , \mathrm { AB } = 2 \mathrm {~m}\) and angle \(\mathrm { OAB } = 120 ^ { \circ }\). Angles \(\theta , \phi\) and \(h\) are as shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-07_949_949_429_214} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(h = \sin \theta + 2 \sin \left( \theta + 60 ^ { \circ } \right)\). The rod is free to rotate about the origin so that \(\theta\) and \(\phi\) vary. You may assume that the result for \(h\) in part (a) holds for all values of \(\theta\).
  2. Find an angle \(\theta\) for which \(h = 0\).
AQA C2 Q6
Moderate -0.8
6
  1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
    1. \(y = 2 \sin x\);
    2. \(y = - \sin x\);
    3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
  2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).
AQA C2 2005 January Q7
11 marks Moderate -0.8
7 The diagram shows the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-4_518_906_1098_552}
  1. Write down the coordinates of the points \(A , B\) and \(C\) marked on the diagram.
  2. Describe the single geometrical transformation by which the curve with equation \(y = \cos 2 x\) can be obtained from the curve with equation \(y = \cos x\).
  3. Solve the equation $$\cos 2 x = 0.37$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). (No credit will be given for simply reading values from a graph.)
    (5 marks)
AQA C2 2006 January Q6
12 marks Moderate -0.8
6
  1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
    1. \(y = 2 \sin x\);
    2. \(y = - \sin x\);
    3. \(y = \sin \left( x - 30 ^ { \circ } \right)\).
  2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).
AQA C2 2008 January Q9
8 marks Standard +0.3
9
  1. Given that $$\frac { 3 + \sin ^ { 2 } \theta } { \cos \theta - 2 } = 3 \cos \theta$$ show that $$\cos \theta = - \frac { 1 } { 2 }$$
  2. Hence solve the equation $$\frac { 3 + \sin ^ { 2 } 3 x } { \cos 3 x - 2 } = 3 \cos 3 x$$ giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
AQA C2 2009 January Q7
13 marks Moderate -0.8
7
  1. Solve the equation \(\sin x = 0.8\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
  2. The diagram shows the graph of the curve \(y = \sin x , 0 \leqslant x \leqslant 2 \pi\) and the lines \(y = k\) and \(y = - k\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line \(y = k\) intersects the curve at the points \(P\) and \(Q\), and the line \(y = - k\) intersects the curve at the points \(R\) and \(S\). The point \(M\) is the minimum point of the curve.
    1. Write down the coordinates of the point \(M\).
    2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of the point \(Q\) in terms of \(\pi\) and \(\alpha\).
    3. Find the length of \(R S\) in terms of \(\pi\) and \(\alpha\), giving your answer in its simplest form.
  3. Sketch the graph of \(y = \sin 2 x\) for \(0 \leqslant x \leqslant 2 \pi\), indicating the coordinates of points where the graph intersects the \(x\)-axis and the coordinates of any maximum points.
AQA C2 2010 January Q8
12 marks Standard +0.3
8
  1. Solve the equation \(\tan \left( x + 52 ^ { \circ } \right) = \tan 22 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Show that the equation $$3 \tan \theta = \frac { 8 } { \sin \theta }$$ can be written as $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
    2. Find the value of \(\cos \theta\) that satisfies the equation $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
    3. Hence solve the equation $$3 \tan 2 x = \frac { 8 } { \sin 2 x }$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
AQA C2 2011 January Q9
10 marks Standard +0.3
9
  1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
    1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
    2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)
AQA C2 2012 January Q8
10 marks Moderate -0.3
8
  1. Given that \(2 \sin \theta = 7 \cos \theta\), find the value of \(\tan \theta\).
    1. Use an appropriate identity to show that the equation $$6 \sin ^ { 2 } x = 4 + \cos x$$ can be written as $$6 \cos ^ { 2 } x + \cos x - 2 = 0$$
    2. Hence solve the equation \(6 \sin ^ { 2 } x = 4 + \cos x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving your answers to the nearest degree.
AQA C2 2013 January Q9
12 marks Standard +0.3
9
  1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    (2 marks)
  2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
    1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
    2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.
AQA C2 2005 June Q8
10 marks Moderate -0.3
8
    1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
    2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
    1. Solve the equation \(4 c ^ { 2 } + 15 c - 4 = 0\).
    2. Hence explain why the only value of \(\cos \theta\) which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is \(\cos \theta = \frac { 1 } { 4 }\).
    3. Hence solve the equation \(4 \tan \theta \sin \theta = 15\) giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Write down all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\) for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.
AQA C2 2006 June Q8
11 marks Moderate -0.3
8
  1. Describe the single geometrical transformation by which the curve with equation \(y = \tan \frac { 1 } { 2 } x\) can be obtained from the curve \(y = \tan x\).
  2. Solve the equation \(\tan \frac { 1 } { 2 } x = 3\) in the interval \(\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }\), giving your answers in radians to three significant figures.
  3. Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval \(0 < \theta < 2 \pi\), giving your answers in radians to three significant figures.
    (5 marks)
AQA C2 2008 June Q9
8 marks Moderate -0.3
9
  1. Solve the equation \(\sin 2 x = \sin 48 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
  2. Solve the equation \(2 \sin \theta - 3 \cos \theta = 0\) in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
AQA C2 2010 June Q7
8 marks Moderate -0.3
7
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
    1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
    2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\), giving your answers in radians to three significant figures.
AQA C2 2012 June Q7
7 marks Moderate -0.3
7 It is given that \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\).
  1. Find the possible values of \(\tan \theta\).
  2. Hence solve the equation \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\), giving all solutions for \(\theta\), in degrees, in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
AQA C2 2013 June Q9
14 marks Standard +0.3
9
    1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
    2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}
AQA C2 2014 June Q7
7 marks Moderate -0.3
7
  1. Given that \(\frac { \cos ^ { 2 } x + 4 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 7\), show that \(\tan ^ { 2 } x = \frac { 3 } { 2 }\).
  2. Hence solve the equation \(\frac { \cos ^ { 2 } 2 \theta + 4 \sin ^ { 2 } 2 \theta } { 1 - \sin ^ { 2 } 2 \theta } = 7\) in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), giving your values of \(\theta\) to the nearest degree.
    [0pt] [4 marks]
AQA C2 2015 June Q6
10 marks Moderate -0.3
6
  1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
  2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
    1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
    2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).
AQA C2 2016 June Q8
9 marks Standard +0.3
8
    1. Given that \(4 \sin x + 5 \cos x = 0\), find the value of \(\tan x\).
    2. Hence solve the equation \(( 1 - \tan x ) ( 4 \sin x + 5 \cos x ) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your values of \(x\) to the nearest degree.
  2. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
    [0pt] [4 marks]
Edexcel C2 Q3
8 marks Moderate -0.3
3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]
Edexcel C2 Q4
10 marks Moderate -0.8
4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$
  1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
  2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
  3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\) [0pt] [P1 June 2002 Question 5]
Edexcel C2 Q3
6 marks Moderate -0.8
3. Giving your answers in terms of \(\pi\), solve the equation $$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
Edexcel C2 Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$
Edexcel C2 Q5
8 marks Standard +0.3
5. (a) Given that $$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0 .$$ (b) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0 .$$
Edexcel C2 Q3
6 marks Moderate -0.8
3. (a) Given that $$5 \cos \theta - 2 \sin \theta = 0 ,$$ show that \(\tan \theta = 2.5\) (b) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.