A-level Further Maths Year 13 · Topic 3
Hyperbolic functions and polar coordinates
Hyperbolic functions behave a lot like trigonometric ones.
We will learn their identities, inverses in logarithmic form and their calculus.
Then we find areas enclosed by polar curves.
- 3 subtopics
- 39 questions
- Free: 3.1 notes and 10 questions
In this topic
- 3.1Hyperbolic functions: identities, equations and inversesFree below · 15 questions
- 3.2Calculus of hyperbolic functionsOpen in Full Marks · 14 questions
- 3.3Areas enclosed by polar curvesOpen in Full Marks · 10 questions
3.1 · Free notes
Hyperbolic functions: identities, equations and inverses
Key points
- cosh x = (eˣ + e⁻ˣ)/2, sinh x = (eˣ − e⁻ˣ)/2, tanh x = sinh x/cosh x; also sech, cosech and coth as reciprocals.
- Identities: cosh²x − sinh²x = 1, sech²x = 1 − tanh²x, sinh 2x = 2sinh x cosh x, cosh 2x = cosh²x + sinh²x = 2cosh²x − 1 = 1 + 2sinh²x.
- Inverses in log form: arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² − 1)) (x ≥ 1); artanh x = ½ ln((1 + x)/(1 − x)) (|x| < 1).
- Solving equations: write in exponentials and solve a quadratic in eˣ, or use identities (for example replace cosh²x by 1 + sinh²x).
In more detail
- Worked example: solve cosh x = 5/4: eˣ + e⁻ˣ = 5/2 → 2e²ˣ − 5eˣ + 2 = 0 → eˣ = 2 or ½ → x = ±ln 2.
- Deriving arsinh: set y = arsinh x, so x = sinh y; then e²ʸ − 2xeʸ − 1 = 0 and eʸ = x + √(x² + 1).
- Exam tip: reject solutions with eˣ ≤ 0, since eˣ is always positive.
Practice
Hyperbolic functions: identities, equations and inverses: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. Which identity is correct?
- cosh 2x = 1 − 2sinh²x
- sinh 2x = 2cosh²x
- cosh 2x = 1 + 2sinh²x
- cosh²x + sinh²x = 1
Show answer
C: cosh 2x = 1 + 2sinh²x. Osborn's rule from cos 2x = 1 − 2sin²x.
2. What is sech x?
- 1/cosh x
- 1/sinh x
- cosh x/sinh x
- sinh x
Show answer
A: 1/cosh x. Reciprocal of cosh.
3. Express arcosh x as a logarithm.
- ln(x + √(x² + 1))
- ½ln((1 + x)/(1 − x))
- ln(x − 1)
- ln(x + √(x² − 1))
Show answer
D: ln(x + √(x² − 1)). x ≥ 1.
4. Solve sinh x = 0.
- x = 1
- x = 0
- x = ±1
- No solution
Show answer
B: x = 0. sinh is zero only at 0.
5. Solve cosh x = 5/4.
- x = ln 2 only
- x = ln(5/4)
- x = ±ln 2
- x = ±1
Show answer
C: x = ±ln 2. ln(5/4 + 3/4) = ln 2.
6. What is the value of artanh 0?
- 0
- 1
- ∞
- Undefined
Show answer
A: 0. ½ ln 1 = 0.
7. Which identity is equivalent to sech²x + tanh²x = …?
- 0
- 2
- cosh 2x
- 1
Show answer
D: 1. Divide cosh²x − sinh²x = 1 by cosh²x.
8. Solve 3sinh x + cosh x = 3 using exponentials. Which quadratic in eˣ results?
- e²ˣ − 3eˣ + 1 = 0
- 2e²ˣ − 3eˣ − 1 = 0
- 4e²ˣ − 6eˣ − 2 = 0
- e²ˣ + 3eˣ − 1 = 0
Show answer
B: 2e²ˣ − 3eˣ − 1 = 0. (3/2)(eˣ − e⁻ˣ) + ½(eˣ + e⁻ˣ) = 3 → 2eˣ − e⁻ˣ = 3.
9. For what x is artanh x defined?
- x ≥ 1
- All x
- |x| < 1
- x > 0
Show answer
C: |x| < 1. The log argument must be positive.
10. How many solutions does cosh x = 3 have?
- Two
- One
- None
- Three
Show answer
A: Two. cosh is even, with minimum 1.
Keep going in Full Marks
Full Marks has 15 questions on Hyperbolic functions: identities, equations and inverses (5 more than here), set out in practice levels. You also get notes and 3D models for all 3 subtopics of Hyperbolic functions and polar coordinates, plus games, timed exams and past papers.
On your exam board
- AQATopic 6: Polar coordinates and hyperbolic functions
- OCR ATopic 4: Polar coordinates and hyperbolic functions
- OCR BTopic 3: Polar coordinates and hyperbolic functions
- EdexcelTopic 4: Polar coordinates, Topic 5: Hyperbolic functions
- CambridgeTopic 1: Hyperbolic functions, Topic 3: Differentiation and integration