Solve using double angle formulas

A question is this type if and only if it asks to solve an equation involving hyperbolic functions where the key step is to use or prove a double angle formula (such as cosh 2x in terms of cosh x or sinh x) before solving algebraically, giving answers in exact logarithmic form.

19 questions · Standard +0.7

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Edexcel F3 2022 January Q1
8 marks Challenging +1.2
1
  1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \cosh ^ { 4 } x = \cosh 4 x + p \cosh 2 x + q$$ where \(p\) and \(q\) are constants to be determined.
  2. Hence, or otherwise, solve the equation $$\cosh 4 x - 17 \cosh 2 x + 9 = 0$$ giving your answers in exact simplified form in terms of natural logarithms.
Edexcel F3 2015 June Q1
7 marks Standard +0.3
  1. Find the exact values of \(x\) for which
$$\cosh 2 x - 7 \sinh x = 5$$ giving your answers as natural logarithms.
Edexcel FP3 2010 June Q3
8 marks Standard +0.3
3. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ (b) Solve the equation $$\cosh 2 x - 3 \sinh x = 15$$ giving your answers as exact logarithms.
Edexcel FP3 2015 June Q1
6 marks Standard +0.3
  1. Solve the equation
$$2 \cosh ^ { 2 } x - 3 \sinh x = 1$$ giving your answers in terms of natural logarithms.
Edexcel FP3 2017 June Q3
9 marks Standard +0.3
3. (a) Using the definition for \(\cosh x\) in terms of exponentials, show that $$\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1$$ (b) Find the exact values of \(x\) for which $$29 \cosh x - 3 \cosh 2 x = 38$$ giving your answers in terms of natural logarithms.
OCR MEI FP2 2006 June Q4
18 marks Challenging +1.2
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
  2. Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
  3. Show that \(\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3\).
  4. Find the exact value of \(\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x\).
OCR FP2 2008 January Q8
10 marks Standard +0.8
8
  1. By using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$\sinh ^ { 3 } x = \frac { 1 } { 4 } \sinh 3 x - \frac { 3 } { 4 } \sinh x$$
  2. Find the range of values of the constant \(k\) for which the equation $$\sinh 3 x = k \sinh x$$ has real solutions other than \(x = 0\).
  3. Given that \(k = 4\), solve the equation in part (ii), giving the non-zero answers in logarithmic form.
OCR FP2 2006 June Q4
7 marks Standard +0.3
4
  1. Using the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), prove that $$\cosh 2 x = 2 \cosh ^ { 2 } x - 1$$
  2. Hence solve the equation $$\cosh 2 x - 7 \cosh x = 3$$ giving your answer in logarithmic form.
OCR FP2 2016 June Q1
9 marks Standard +0.8
1
  1. By first expanding \(\left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 3 }\), or otherwise, show that \(\cosh 3 x \equiv 4 \cosh ^ { 3 } x - 3 \cosh x\).
  2. Solve the equation \(\cosh 3 x = 6 \cosh x\), giving your answers in exact logarithmic form.
OCR MEI FP2 2008 January Q4
18 marks Standard +0.8
4
  1. Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
  3. Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
  4. Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .
OCR MEI FP2 2010 January Q4
18 marks Standard +0.8
4
  1. Prove, using exponential functions, that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ Differentiate this result to obtain a formula for \(\sinh 2 x\).
  2. Solve the equation $$2 \cosh 2 x + 3 \sinh x = 3$$ expressing your answers in exact logarithmic form.
  3. Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\). Find the exact value of the integral $$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$
OCR MEI FP2 2011 June Q4
18 marks Challenging +1.2
4
  1. Given that \(\cosh y = x\), show that \(y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\) and that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x\), expressing your answer in an exact logarithmic form.
  3. Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.
OCR FP2 2011 January Q4
9 marks Standard +0.8
4
  1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \sinh ^ { 4 } x \equiv \cosh 4 x - 4 \cosh 2 x + 3$$
  2. Solve the equation $$\cosh 4 x - 3 \cosh 2 x + 1 = 0$$ giving your answer(s) in logarithmic form.
OCR FP2 2012 June Q3
8 marks Standard +0.3
3
  1. By quoting results given in the List of Formulae (MF1), prove that \(\tanh 2 x \equiv \frac { 2 \tanh x } { 1 + \tanh ^ { 2 } x }\).
  2. Solve the equation \(5 \tanh 2 x = 1 + 6 \tanh x\), giving your answers in logarithmic form.
OCR Further Pure Core 2 2022 June Q5
7 marks Standard +0.3
5
  1. By using the exponential definitions of \(\sinh x\) and \(\cosh x\), prove the identity \(\cosh 2 x \equiv \cosh ^ { 2 } x + \sinh ^ { 2 } x\).
  2. Hence find an expression for \(\cosh 2 x\) in terms of \(\cosh x\).
  3. Determine the solutions of the equation \(5 \cosh 2 x = 16 \cosh x + 21\), giving your answers in exact logarithmic form.
OCR Further Pure Core 2 2021 November Q5
8 marks Standard +0.8
5 In this question you must show detailed reasoning.
  1. Using the definition of \(\cosh x\) in terms of exponentials, show that \(\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1\).
  2. Solve the equation \(\cosh 2 x = 3 \cosh x + 1\), giving all your answers in exact logarithmic form.
OCR Further Pure Core 2 2023 June Q5
7 marks Challenging +1.2
5 In this question you must show detailed reasoning.
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that \(\sinh 2 x \equiv 2 \sinh x \cosh x\).
  2. Solve the equation \(15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20\), giving all your answers in logarithmic form.
AQA FP2 2009 January Q1
9 marks Standard +0.3
1
  1. Use the definitions \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\) and \(\cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)\) to show that $$1 + 2 \sinh ^ { 2 } \theta = \cosh 2 \theta$$
  2. Solve the equation $$3 \cosh 2 \theta = 2 \sinh \theta + 11$$ giving each of your answers in the form \(\ln p\).
AQA Further AS Paper 1 Specimen Q6
12 marks Challenging +1.2
6
  1. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \frac { 1 } { 2 } \ln \left( \frac { 1 + t } { 1 - t } \right)\) where \(t = \tanh x\)
    [0pt] [4 marks]
    6
    1. Prove \(\cosh ^ { 3 } x = \frac { 1 } { 4 } \cosh 3 x + \frac { 3 } { 4 } \cosh x\)
      [0pt] [4 marks] 6
  3. (ii) Show that the equation \(\cosh 3 x = 13 \cosh x\) has only one positive solution.
    Find this solution in exact logarithmic form.
    [0pt] [4 marks]