Solve double/multiple angle equation

A question is this type if and only if it requires solving trig(nx) = k for n ≥ 2 with no phase shift, by expanding the range of the substituted angle and listing all solutions.

9 questions · Moderate -0.5

1.05o Trigonometric equations: solve in given intervals
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CAIE P1 2016 November Q10
11 marks Standard +0.3
10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
  4. State the largest value of \(k\) for which g has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
OCR C2 Q8
11 marks Standard +0.3
8. (i) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\tan 2 x = 3$$ (ii) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y$$
OCR MEI C2 Q10
5 marks Moderate -0.8
10
  1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
  2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR C2 2011 January Q7
8 marks Moderate -0.3
7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(3 \tan 2 x = 1\)
  2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)
OCR C2 2013 June Q2
6 marks Moderate -0.3
2 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. \(\sin \frac { 1 } { 2 } x = 0.8\)
  2. \(\sin x = 3 \cos x\)
OCR MEI Paper 2 2024 June Q4
5 marks Easy -1.2
4
  1. On the axes in the Printed Answer Booklet, sketch the graph of \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve the equation \(\sin 2 \theta = - \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\). \(5 M\) is the event that an A-level student selected at random studies mathematics. \(C\) is the event that an A-level student selected at random studies chemistry.
    You are given that \(\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36\) and \(\mathrm { P } ( \mathrm { M }\) and \(\mathrm { C } ) = 0.24\). These probabilities are shown in the two-way table below.
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(M\)\(M ^ { \prime }\)Total
    \(C\)0.240.36
    \(C ^ { \prime }\)
    Total0.421
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)
Edexcel C2 Q6
9 marks Moderate -0.8
6. $$f ( x ) = \cos 2 x , \quad 0 \leq x \leq \pi .$$
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
  3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).
AQA C2 2007 January Q8
12 marks Moderate -0.8
8
  1. Solve the equation \(\cos x = 0.3\) 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 \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). 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 \(Q\) in terms of \(\pi\) and \(\alpha\).
  3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
  4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
    (4 marks)