Hyperbolic Functions

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

A-Level Further Mathematics revision: Hyperbolic Functions. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.

Exam Board:

📌 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

📚 Lesson Plan (50 minutes)

Starter (5 min): Recall prior knowledge of hyperbolic functions with quick questions.
Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
Key points review (5 min): Revisit the key points together, confirming understanding.
Worked example (10 min): Model the example question: Solve 2 cosh x - 3 sinh x = 1. Solution: 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 (10 min): Students attempt the practice questions independently; circulate and support.
Plenary (5 min): Review answers and address misconceptions.

🏠 Homework

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

🧾 Assessment

Check practice answers against the model answer; use the built-in practice questions as formative assessment.

Topics