Functions built from exponentials that behave remarkably like the trigonometric ones, with the signs of a few identities flipped. They arise naturally in integration.
Syllabus points covered
- Understand the definitions of sinh x, cosh x and tanh x in terms of exponentials and sketch their graphs
- Prove and use hyperbolic identities, including cosh²x − sinh²x = 1
- Understand and use the inverse hyperbolic functions and their logarithmic forms
- Solve equations involving hyperbolic functions
Common mistakes
- Using the trigonometric identity sin² + cos² = 1 instead of the hyperbolic one, where the sign differs.
- Assuming cosh x can be negative — it cannot.
- Forgetting that the inverse of cosh needs a restricted domain to be a function.
- Losing a solution when solving a hyperbolic equation that reduces to a quadratic in eˣ.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheetComing soon
- Worksheet answersComing soon
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers