4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

12 The equation \(x ^ { 3 } - 2 x ^ { 2 } - x + 2 = 0\) has three roots. One of the roots is 2 12

1. Find the other two roots of the equation. 12
2. Hence, or otherwise, solve $$\cosh ^ { 3 } \theta - 2 \cosh ^ { 2 } \theta - \cosh \theta + 2 = 0$$ giving your answers in an exact form.

1 Which of the following exponential expressions is equivalent to \(2 \sinh x\) ?
Circle your answer. \(\mathrm { e } ^ { x }\) \(\mathrm { e } ^ { x } + \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { x } - \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { - x }\)

15 The two values of \(\theta\) that satisfy the equation $$\sinh ^ { 2 } \theta - \sinh \theta - 2 = 0$$ are \(\theta _ { 1 }\) and \(\theta _ { 2 }\) 15

1. Hamzah is asked to find the value of \(\theta _ { 1 } + \theta _ { 2 }\) He writes his answer as follows:
   The quadratic coefficients are \(a = 1 , b = - 1 , c = - 2\) The sum of the roots is \(- \frac { b } { a }\) So \(\theta _ { 1 } + \theta _ { 2 } = - \frac { - 1 } { 1 } = 1\) Explain Hamzah's error.
   [0pt] [1 mark] 15
2. Find the correct value of \(\theta _ { 1 } + \theta _ { 2 }\) Give your answer as a single logarithm. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-28_2492_1721_217_150}

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
2. (i) Prove \(\cosh ^ { 3 } x = \frac { 1 } { 4 } \cosh 3 x + \frac { 3 } { 4 } \cosh x\) [0pt] [4 marks] 6 (b) (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]

14

1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}

12 The integral \(S _ { n }\) is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12

1. Show that for \(n \geq 2\) $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$

   12	Hence show that \(\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24\)

8

1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
   1. (i) Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\) 8
   2. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\) 8
   3. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$

3 The diagram shows part of the curve \(y = 5 \cosh x + 3 \sinh x\). \includegraphics[max width=\textwidth, alt={}, center]{ef967953-70b5-4dd1-a342-ad488b5fa79f-02_426_661_906_260}

1. Solve the equation \(5 \cosh x + 3 \sinh x = 4\) giving your solution in exact form.
2. In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x\) giving your answer in the form \(a \mathrm { e } + \frac { b } { \mathrm { e } }\) where \(a\) and \(b\) are integers to be determined.

5 The function \(\operatorname { sech } x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).

1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).

3

1. Using exponentials, show that \(\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1\).
2. By differentiating both sides of the identity in part (a) with respect to \(u\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\).
3. Use the substitution \(x = \sinh ^ { 2 } u\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form \(a \sinh ^ { - 1 } b \sqrt { x } + \mathrm { f } ( x )\) where \(a\) and \(b\) are integers and \(\mathrm { f } ( x )\) is a function to be determined.
4. Hence determine the exact area of the region between the curve \(y = \sqrt { \frac { x } { x + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(p + q \ln r\) where \(p , q\) and \(r\) are numbers to be determined.

2 In this question you must show detailed reasoning.
Solve the equation \(2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0\) giving each answer in the form \(\ln ( p + q \sqrt { r } )\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.

1. State the plane of reflection of \(R\).
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
   1. By using Euler's formula show that \(\cosh ( \mathrm { iz } ) = \cos z\).
   2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
   3. 1. Write down the general solution to (\textit{).
      2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where \(\lambda\) is a positive constant.
   4. In the case where \(\lambda = 16\) the door is held open at an angle of 0.9 radians and then released from rest at time \(t = 0\).
      1. Find, in a real form, the general solution of ( \(\dagger\) ).
      2. Find the particular solution of ( \(\dagger\) ).
      3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( \(\dagger\) ) improves the model.
      4. Find the value of \(\lambda\) for which the door is critically damped.

4

1. Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
2. Solve the equation \(5 \cosh x + 3 \sinh x = 12\), giving your answers in the form \(\ln ( p \pm q \sqrt { 2 } )\) for rational numbers \(p\) and \(q\) to be determined.

12

1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
   By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
   1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
   2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
   3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
   4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$

13

1. Use the definitions \(\tanh \theta = \frac { \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) and \(\operatorname { sech } \theta = \frac { 2 } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) to prove the results
   1. \(\tanh ^ { 2 } \theta \equiv 1 - \operatorname { sech } ^ { 2 } \theta\),
   2. \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tanh \theta ) = \operatorname { sech } ^ { 2 } \theta\).
   3. Let \(I _ { n } = \int _ { 0 } ^ { \alpha } \tanh ^ { 2 n } \theta \mathrm {~d} \theta\) for \(n \geqslant 0\), where \(\alpha > 0\).
      (a) Show that \(I _ { n - 1 } - I _ { n } = \frac { \tanh ^ { 2 n - 1 } \alpha } { 2 n - 1 }\) for \(n \geqslant 1\). Given that \(\alpha = \frac { 1 } { 2 } \ln 3\),
      (b) evaluate \(I _ { 0 }\),
   4. use the method of differences to show that \(I _ { n } = \frac { 1 } { 2 } \ln 3 - \sum _ { r = 1 } ^ { n } \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 r - 1 } } { 2 r - 1 }\) and deduce the sum of the infinite series \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).

12

1. (a) Show that \(\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }\).
   (b) Hence, or otherwise, show that, if \(\tanh x = \frac { 1 } { k }\) for \(k > 1\), then \(x = \frac { 1 } { 2 } \ln \left( \frac { k + 1 } { k - 1 } \right)\) and find an expression in terms of \(k\) for \(\sinh 2 x\).
2. A curve has equation \(y = \frac { 1 } { 2 } \ln ( \tanh x )\) for \(\alpha \leqslant x \leqslant \beta\), where \(\tanh \alpha = \frac { 1 } { 3 }\) and \(\tanh \beta = \frac { 1 } { 2 }\). Find, in its simplest exact form, the arc length of this curve.

6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).

1. Show that \(2.5 < \alpha < 3\).
2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.

12 The curve \(C\) has equation \(y = \ln \left( \tanh \frac { 1 } { 2 } x \right)\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x\).
2. For positive integers \(n\), the length of the arc of \(C\) between \(x = n\) and \(x = 2 n\) is \(L _ { n }\).
   1. Show by calculus that, when \(n\) is large, \(L _ { n } \approx n\).
   2. Explain how this result corresponds to the shape of \(C\).

5

1. Use the definition \(\tanh y = \frac { \mathrm { e } ^ { 2 y } - 1 } { \mathrm { e } ^ { 2 y } + 1 }\) to show that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\) for \(| x | < 1\).
2. Solve the equation \(\tanh x + \operatorname { coth } x = 4\), giving your answer in the form \(p \ln m\), where \(p\) is a positive rational number and \(m\) is a positive integer.

3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.

12 The curve \(C\) is defined parametrically by $$x = t + \ln ( \cosh t ) , \quad y = \sinh t$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \cosh ^ { 2 } t\).
2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \cosh ^ { 2 } t ( 2 \sinh t - \cosh t )\).
3. Find the exact value of \(t\) at the point on \(C\) where \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

It is given that \(y = \sinh(x^2) + \cosh(x^2)\).

1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]

The curve \(C\) has equation $$y = 9 \cosh x + 3 \sinh x + 7x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm. [6]