4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1

16

1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Determine the values of \(x\) for which $$64 \cosh ^ { 4 } x - 64 \cosh ^ { 2 } x - 9 = 0$$ Give your answers in the form \(q \ln 2\) where \(q\) is rational and in simplest form.

1. The curve \(C\) has equation

$$y = 31 \sinh x - 2 \sinh 2 x \quad x \in \mathbb { R }$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\).

1. (a) Using the definition of \(\sinh x\) in terms of exponentials, prove that

$$4 \sinh ^ { 3 } x + 3 \sinh x \equiv \sinh 3 x$$ (b) Hence solve the equation $$\sinh 3 x = 19 \sinh x$$ giving your answers as simplified natural logarithms where appropriate.

5. \begin{figure}[h] \end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.

1. Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\). The length of the curved cross-section of the miniature arch is required to be 2 m long.
2. Find the height of the arch, according to this model, giving your answer to 2 significant figures.
3. Find also the width of the base of the arch giving your answer to 2 significant figures.
4. Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
   (You do not need to consider the arc length of your curve)

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.

7

1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$ \section*{(ii) In this question you must show detailed reasoning.} By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3 .$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\), where \(a\) is a rational number.

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 \(\ln p\).

4

1. Sketch the graph of \(y = \tanh x\).
2. Given that \(u = \tanh x\), use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
3. 1. Show that the equation $$3 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 5$$ can be written as $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$
   2. Show that the equation $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$ has only one solution for \(x\).
      Find this solution in the form \(\frac { 1 } { 2 } \ln a\), where \(a\) is an integer.

2

1. Sketch the graph of \(y = \tanh x\) and state the equations of its asymptotes.
2. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
3. Solve the equation \(6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x\), giving your answers in terms of natural logarithms.
   [0pt] [5 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{bc3aaed2-4aef-4aec-b657-098b1e581e55-04_855_1447_920_324}

6 Prove the identity $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$

1 Express \(\cosh ^ { 2 } x\) in terms of \(\sinh x\) Circle your answer. \(1 + \sinh ^ { 2 } x\) \(1 - \sinh ^ { 2 } x\) \(\sinh ^ { 2 } x - 1\) \(- 1 - \sinh ^ { 2 } x\)

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}

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 \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).

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

1. Starting from the definitions of \(\tanh\) and \(\sech\) in terms of exponentials, prove that $$1 - \tanh^2 \theta = \sech^2 \theta.$$ [3]

The variables \(x\) and \(y\) are such that \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\), for \(-\frac{1}{4}\pi < x < \frac{3}{4}\pi\).

1. By differentiating the equation \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\) with respect to \(x\), show that $$\frac{dy}{dx} = -\operatorname{cosec}\left(x + \frac{1}{4}\pi\right).$$ [4]
2. Hence find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\cos\left(x + \frac{1}{4}\pi\right)\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [5]

The hyperbola \(H\) has equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

1. Use calculus to show that the equation of the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\) may be written in the form $$xb\cosh\theta - ya\sinh\theta = ab$$ [4] The line \(l_1\) is the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\), \(\theta \neq 0\). Given that \(l_1\) meets the \(x\)-axis at the point \(P\),
2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). [2] The line \(l_2\) is the tangent to \(H\) at the point \((a, 0)\). Given that \(l_1\) and \(l_2\) meet at the point \(Q\),
3. find, in terms of \(a\), \(b\) and \(\theta\), the coordinates of \(Q\). [2]
4. Show that, as \(\theta\) varies, the locus of the mid-point of \(PQ\) has equation $$x(4y^2 + b^2) = ab^2$$ [6]

Using the definitions of hyperbolic functions in terms of exponentials,

1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]