Hyperbolic functions and their inverses
Define hyperbolic functions from exponentials, use their identities and graphs, and work with inverse hyperbolic functions.
Section Paper 2 · Further Pure Mathematics 2 · Topics 2.1–2.6
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Use definitions of sinh, cosh, tanh, sech, cosech and coth
- Sketch hyperbolic graphs and prove or use identities
- Use inverse hyperbolic functions and their logarithmic forms
Exponentials define the hyperbolic functions
sinh x=(eˣ−e⁻ˣ)/2 and cosh x=(eˣ+e⁻ˣ)/2. Then tanh x=sinh x/cosh x, sech x=1/cosh x, cosech x=1/sinh x and coth x=1/tanh x where defined.
The key identity is cosh²x−sinh²x=1. Unlike sin and cos, cosh²+sinh² is not a constant. cosh is even with minimum 1; sinh and tanh are odd; tanh approaches −1 and 1 as x tends to −∞ and +∞.
Inverse functions require domain checks
The inverse hyperbolic functions undo the corresponding one-to-one branches. For example, if y=sinh⁻¹x, then sinh y=x and y=ln(x+√(x²+1)).
For cosh⁻¹x, the real domain is x≥1 and cosh⁻¹x=ln(x+√(x²−1)). For tanh⁻¹x, the real domain is |x|<1 and tanh⁻¹x=½ln((1+x)/(1−x)).
Use the stated real domain before applying a logarithmic form; an algebraic rearrangement can introduce an invalid branch.
Pause and explain
Which habit best checks your work on hyperbolic functions and their inverses?
Worked example
Solve cosh x=5/3 for real x.
Show the worked solution
- Use cosh x=(eˣ+e⁻ˣ)/2 and set u=eˣ>0.
- Then u+1/u=10/3, so 3u²−10u+3=0 and (3u−1)(u−3)=0.
- Thus u=3 or u=1/3; because u=eˣ, x=ln 3 or x=−ln 3.
Answer x=±ln 3.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Given sinh x=3/4 and x>0, find cosh x and tanh x.
Give me a hint
Use cosh²x−sinh²x=1 and choose the positive cosh value.
Compare my reasoning
- cosh²x=1+9/16=25/16.
- cosh x=5/4 because cosh is positive for real x.
- tanh x=(3/4)/(5/4).
cosh x=5/4 and tanh x=3/5.
Look for these in your work
- Identity applied
- Positive branch chosen
- Ratio simplified
Solve sinh x=2 for real x, giving an exact logarithmic answer.
Give me a hint
Use the inverse function formula or set u=eˣ and use u>0.
Compare my reasoning
- x=sinh⁻¹2=ln(2+√(2²+1)).
- Simplify the square root to √5.
- The logarithm's argument is positive, so this gives a real solution.
x=ln(2+√5).
Look for these in your work
- Inverse formula used
- Radical simplified
- Real branch confirmed
A learner claims tanh x=1 for a finite real x because tanh x approaches 1 for large x. Explain the error.
Give me a hint
Compare a limit with an attained value using tanh x=(e²ˣ−1)/(e²ˣ+1).
Compare my reasoning
- For any finite x, e²ˣ is positive and finite.
- The numerator is smaller than the denominator, so tanh x<1.
- The value 1 is a horizontal asymptote approached as x→∞, not an attained value.
tanh x approaches 1 but never equals 1 at finite real x.
Look for these in your work
- Finite exponential considered
- Strict inequality justified
- Limit distinguished from value
Common mistakes
- Using a trigonometric identity with hyperbolic functions
- Forgetting cosh x is even and therefore not one-to-one on all real numbers
- Using a real inverse-hyperbolic formula outside its domain
What can you explain now?
Which identity relates cosh²x and sinh²x?
Compare with the explanation
cosh²x−sinh²x=1
It follows directly from the exponential definitions.
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Make it stick
How do the exponential definitions determine the main hyperbolic identity, graph symmetry and inverse domains?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.