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

8

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 \(\mathrm { x } = \sinh ^ { 2 } \mathrm { u }\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form asinh \(^ { - 1 } \mathrm {~b} \sqrt { \mathrm { x } } + \mathrm { f } ( \mathrm { x } )\) 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 \(\mathrm { y } = \sqrt { \frac { \mathrm { x } } { \mathrm { x } + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(\mathrm { p } + \mathrm { q } \mid \mathrm { nr }\) where \(p , q\) and \(r\) are numbers to be determined.

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.

9 Two thin poles, \(O A\) and \(B C\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \(( 0,3 ) , ( 5,0 )\) and \(( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{c07ba83a-75fa-42dc-9bfd-6fc2f9226a23-5_805_1554_452_258} It is required to find the height of pole \(B C\) by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation \(\mathrm { y } = \mathrm { k } \cosh ( \mathrm { ax } - \mathrm { b } ) - 1\) where \(k , a\) and \(b\) are positive constants.

1. Determine the value of \(k\).
2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. Holly models the curve using the equation \(y = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 3\).
3. Write down the coordinates of the point, \(( u , v )\) where \(u\) and \(v\) are both non-zero, at which the two models will agree.
4. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(B C\) by 3.32 m to 3 significant figures.

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.

5 A designer intends to manufacture a product using a 3-D printer. The product will take the form of a surface \(S\) which must meet a number of design specifications. The designer chooses to model \(S\) with the equation \(\mathrm { Z } = \mathrm { y } \cosh \mathrm { x }\) for \(- \ln 20 \leqslant x \leqslant \ln 20 , - 2 \leqslant y \leqslant 2\).

1. 1. In the Printed Answer Booklet, on the axes provided, sketch the section of \(S\) given by \(y = 1\).
   2. One of the design specifications of the product is that this section should have a length no greater than 20 units. Determine whether the product meets this requirement according to the model.
2. 1. In the Printed Answer Booklet, on the axes provided, sketch the contour of \(S\) given by \(z = 1\).
   2. When this contour is rotated through \(2 \pi\) radians about the \(x\)-axis, the surface \(T\) is generated. The surface area of \(T\) is denoted by \(A\). Show that \(A\) can be written in the form \(\mathrm { k } \pi \int _ { 0 } ^ { \ln 20 } \frac { 1 } { \cosh ^ { 3 } \mathrm { x } } \sqrt { \cosh ^ { 4 } \mathrm { x } + \cosh ^ { 2 } \mathrm { x } - 1 } \mathrm { dx }\) for some
      integer \(k\) to be determined. integer \(k\) to be determined.
   3. A second design specification is that the surface area of \(T\) must not be greater than 20 square units. Use your calculator to decide whether the product meets this requirement according to the model.

1

1. Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
2. 1. Express $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$ in terms of \(\cosh x\).
   2. Sketch the curve \(y = \cosh x\).
   3. Hence solve the equation $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$ giving your answers in logarithmic form.

4 A curve \(C\) is given parametrically by the equations $$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$

1. Express $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$ in terms of \(\cosh t\).
2. The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that \(S\), the area of the curved surface generated, is given by $$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
   2. Find the exact value of \(S\).

4

1. Prove that the curve $$y = 12 \cosh x - 8 \sinh x - x$$ has exactly one stationary point.
2. Given that the coordinates of this stationary point are \(( a , b )\), show that \(a + b = 9\).

1

1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).

3 A curve has cartesian equation $$y = \frac { 1 } { 2 } \ln ( \tanh x )$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sinh 2 x }$$
2. The points \(A\) and \(B\) on the curve have \(x\)-coordinates \(\ln 2\) and \(\ln 4\) respectively. Find the arc length \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.

1

1. Express $$5 \sinh x + \cosh x$$ in the form \(A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }\), where \(A\) and \(B\) are integers.
2. Solve the equation $$5 \sinh x + \cosh x + 5 = 0$$ giving your answer in the form \(\ln a\), where \(a\) is a rational number.

5

1. Use the definition \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) to show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
   (2 marks)
2. 1. The arc of the curve \(y = \cosh x\) between \(x = 0\) and \(x = \ln a\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that \(S\), the surface area generated, is given by $$S = 2 \pi \int _ { 0 } ^ { \ln a } \cosh ^ { 2 } x \mathrm {~d} x$$
   2. Hence show that $$S = \pi \left( \ln a + \frac { a ^ { 4 } - 1 } { 4 a ^ { 2 } } \right)$$

1

1. Show that $$9 \sinh x - \cosh x = 4 \mathrm { e } ^ { x } - 5 \mathrm { e } ^ { - x }$$
2. Given that $$9 \sinh x - \cosh x = 8$$ find the exact value of \(\tanh x\).

1

1. Sketch the curve \(y = \cosh x\).
2. Solve the equation $$6 \cosh ^ { 2 } x - 7 \cosh x - 5 = 0$$ giving your answers in logarithmic form.

2

1. 1. Sketch on the axes below the graphs of \(y = \sinh x\) and \(y = \cosh x\).
   2. Use your graphs to explain why the equation $$( k + \sinh x ) \cosh x = 0$$ where \(k\) is a constant, has exactly one solution.
2. A curve \(C\) has equation \(y = 6 \sinh x + \cosh ^ { 2 } x\). Show that \(C\) has only one stationary point and show that its \(y\)-coordinate is an integer. \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_704_1416_171} \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_711_1416_964}

6

1. Show that \(\frac { 1 } { 5 \cosh x - 3 \sinh x } = \frac { \mathrm { e } ^ { x } } { m + \mathrm { e } ^ { 2 x } }\), where \(m\) is an integer.
2. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$\int _ { 0 } ^ { \ln 2 } \frac { 1 } { 5 \cosh x - 3 \sinh x } \mathrm {~d} x = \frac { \pi } { 8 } - \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$$

5

1. Using the definition \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\), prove the identity $$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
2. Given that \(x = \sinh \theta\) and \(16 x ^ { 3 } + 12 x - 3 = 0\), find the value of \(\theta\) in terms of a natural logarithm.
3. Hence find the real root of the equation \(16 x ^ { 3 } + 12 x - 3 = 0\), giving your answer in the form \(2 ^ { p } - 2 ^ { q }\), where \(p\) and \(q\) are rational numbers.
   [0pt] [2 marks]

3 In this question you must show detailed reasoning.
Solve the equation \(3 \cosh x = 2 \sinh ^ { 2 } x\), giving your solutions in exact logarithmic form.

9 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \ln ( 1 + \sinh x )\).

1. Given that \(k\) lies in the domain of this function, explain why \(k\) must be greater than \(\ln ( \sqrt { 2 } - 1 )\).
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Show that \(\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) = \frac { \mathrm { a } \sinh \mathrm { x } + \mathrm { b } } { ( 1 + \sinh \mathrm { x } ) ^ { 2 } }\), where \(a\) and \(b\) are integers to be determined.
3. Hence find a quadratic approximation to \(\mathrm { f } ( x )\) for small values of \(x\).
4. Find the percentage error in this approximation when \(x = 0.1\).

12 The diagram shows the curve with parametric equations \(x = 2 \cosh t + \sinh t , y = \cosh t - 2 \sinh t\). \includegraphics[max width=\textwidth, alt={}, center]{83275e7c-7f5a-4f26-b81d-a041e67ac9a2-5_812_808_1283_246}

1. The curve crosses the positive \(x\)-axis at A .
   1. Determine the value of the parameter \(t\) at A , giving your answer in logarithmic form.
   2. Find the \(x\)-coordinate of A , giving your answer correct to \(\mathbf { 3 }\) significant figures.
2. The point B has parameter \(t = 0\). Determine the equation of the tangent to the curve at B .

13

1. Using exponentials, prove that \(\sinh 2 x = 2 \cosh x \sinh x\).
2. Hence show that if \(\mathrm { f } ( x ) = \sinh ^ { 2 } x\), then \(\mathrm { f } ^ { \prime \prime } ( x ) = 2 \cosh 2 x\).
3. Explain why the coefficients of odd powers in the Maclaurin series for \(\sinh ^ { 2 } x\) are all zero.
4. Find the coefficient of \(x ^ { n }\) in this series when \(n\) is a positive even number.

3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below. The formula in cell D5 is \(\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2\).

1. Write down the equation which is being solved. The formula in cell C 6 is \(\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )\).
2. Write down a similar formula for cell E6.
3. Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
4. Explain what the values in column J tell you about

1. (i) (a) Explain why \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is an improper integral.
   (b) Show that \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is divergent.
   (ii)

$$4 \sinh x = p \cosh x \quad \text { where } p \text { is a real constant }$$ Given that this equation has real solutions, determine the range of possible values for \(p\)

1. (a) Prove that

$$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$ stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation $$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$

9. $$y = \cosh ^ { n } x \quad n \geqslant 5$$

1. 1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
   2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.