Express in form R cosh(x±α) or R sinh(x±α)

1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that

$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)

1. (a) Use the definitions of hyperbolic functions in terms of exponentials to show that

$$\sinh ( A + B ) \equiv \sinh A \cosh B + \cosh A \sinh B$$ (b) Hence express \(10 \sinh x + 8 \cosh x\) in the form \(R \sinh ( x + \alpha )\) where \(R > 0\), giving \(\alpha\) in the form \(\ln p\) where \(p\) is an integer.
(c) Hence solve the equation $$10 \sinh x + 8 \cosh x = 18 \sqrt { 7 }$$ giving your answer in the form \(\ln ( \sqrt { 7 } + q )\) where \(q\) is a rational number to be determined.

8

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
   (a) \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
   (b) \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
2. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
3. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
4. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).

11. (a) Show that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , \quad \text { where } - 1 < x < 1$$ (b) Given that $$a \cosh x + b \sinh x \equiv \operatorname { rcosh } ( x + \alpha ) , \quad \text { where } a > b > 0$$ show that $$\alpha = \frac { 1 } { 2 } \ln \left( \frac { a + b } { a - b } \right)$$ and find an expression for \(r\) in terms of \(a\) and \(b\).
(c) Hence solve the equation $$5 \cosh x + 4 \sinh x = 10$$ giving your answers correct to three significant figures.