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

A question is this type if and only if it asks to express a linear combination a cosh x + b sinh x in the form R cosh(x + α) or R sinh(x + α) using addition formulas.

4 questions · Standard +0.6

4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1
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Edexcel F3 2022 June Q1
7 marks Standard +0.3
  1. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
  2. 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)
Edexcel F3 2024 June Q4
9 marks Standard +0.3
  1. Use the definitions of hyperbolic functions in terms of exponentials to show that $$\sinh ( A + B ) \equiv \sinh A \cosh B + \cosh A \sinh B$$
  2. 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.
  3. 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.
OCR FP2 2009 June Q8
14 marks Standard +0.3
8
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
    1. \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
    2. \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
    3. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
    4. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
    5. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).
WJEC Further Unit 4 Specimen Q11
17 marks Challenging +1.3
  1. Show that $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right), \quad \text{where } -1 < x < 1.$$ [4]
  2. Given that $$a \cosh x + b \sinh x \equiv r \cosh(x + \alpha), \quad \text{where } a > b > 0,$$ 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 \(b\). [7]
  3. Hence solve the equation $$5 \cosh x + 4 \sinh x = 10,$$ giving your answers correct to three significant figures. [6]