amazon.co.ukHyperbolic Functions
A-Level Further Mathematics revision: Hyperbolic Functions. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x/cosh x
Key Fact: Identities: cosh^2x - sinh^2x = 1, 1 - tanh^2x = sech^2x, sinh 2x = 2 sinh x cosh x, cosh 2x = cosh^2x + sinh^2x
Key Fact: Derivatives: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech^2x
Key Fact: Integrals: ∫sinh x dx = cosh x + C, ∫cosh x dx = sinh x + C, ∫sech^2x dx = tanh x + C
Key Fact: Inverse functions: arsinh x = ln(x + sqrt(x^2+1)), arcosh x = ln(x + sqrt(x^2-1)) (x>=1), artanh x = ½ ln((1+x)/(1-x)) (|x|<1)
Key Fact: Osborn's rule: trig identity -> hyperbolic by changing sign of sin^2 terms
🎯 Learning Objectives
- Define sinh, cosh, tanh, sech, cosech, coth in terms of exponentials
- Prove and use hyperbolic identities (analogous to trig identities)
- Differentiate and integrate hyperbolic functions
- Solve equations involving hyperbolic functions
- Use inverse hyperbolic functions and their logarithmic forms
- Apply hyperbolic functions to calculus problems (e.g. catenary)
💡 Worked Example
Exam-Style Question
Question: Solve 2 cosh x - 3 sinh x = 1
Model Answer:
2(eˣ+e⁻ˣ)/2 - 3(eˣ-e⁻ˣ)/2 = 1 -> eˣ+e⁻ˣ - 1.5(eˣ-e⁻ˣ) = 1 -> -0.5eˣ + 2.5e⁻ˣ = 1 -> multiply 2eˣ: -e^2ˣ + 5 = 2eˣ -> e^2ˣ + 2eˣ - 5 = 0 -> eˣ = -1 +/- sqrt6 -> eˣ = sqrt6 - 1 -> x = ln(sqrt6 - 1)
❓ Practice Questions
Questions:
- Prove: cosh 2x = 2cosh^2x - 1 = 1 + 2sinh^2x
- ∫ sinh^2x dx
- Solve: tanh x = ½
- Find arsinh(2)
- Show that the catenary y = cosh x has arc length ∫cosh x dx from a to b
🎬 Video Resources
- Hyperbolic Functions - Overview -- YouTube Search
- Hyperbolic Functions - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Hyperbolic Functions -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Hyperbolic Functions -- Free resources, slides, and interactive questions
- BBC Bitesize - Hyperbolic Functions -- BBC Bitesize revision resources