Solve using sech/tanh identities

1. Solve the equation

$$4 \tanh x - \operatorname { sech } x = 1$$ giving your answer in the form \(x = \ln k\) where \(k\) is a fully simplified rational number.
(6)

1. Solve the equation

$$15 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 13$$ Give your answers in terms of simplified natural logarithms.

1. Find the exact values of x for which

$$4 \tanh ^ { 2 } x - 2 \operatorname { sech } ^ { 2 } x = 3 ,$$ giving your answers in the form \(\pm \ln \mathrm { a }\), where a is real.

1. (a) Using the definitions of hyperbolic functions in terms of exponentials, show that

$$1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$$ (b) Solve the equation $$2 \operatorname { sech } ^ { 2 } x + 3 \tanh x = 3$$ giving your answer as an exact logarithm.
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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. Solve the equation

$$5 \tanh x + 7 = 5 \operatorname { sech } x$$ Give each answer in the form \(\ln k\) where \(k\) is a rational number.

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.

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}
4. 
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4 Show that the solutions to the equation $$3 \tanh ^ { 2 } x - 2 \operatorname { sech } x = 2$$ can be expressed in the form $$x = \pm \ln ( a + \sqrt { b } )$$ where \(a\) and \(b\) are integers to be found.
You may use without proof the result \(\cosh ^ { - 1 } y = \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right)\)

6

1. Given that \(| x | < 1\), prove that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$ 6
2. Solve the equation $$20 \operatorname { sech } ^ { 2 } x - 11 \tanh x = 16$$ Give your answer in logarithmic form.
   \(7 \quad\) The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 7 & - 3 \\ 3 & 6 & k + 1 \\ 1 & 3 & 2 \end{array} \right]$$ where \(k\) is a constant.