All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

Hyperbolic functions: definitions and graphs

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

The hyperbolic functions are built from e^x and e^-x. They behave a lot like sin and cos, but their graphs and some signs are different. Everything in F3 starts from these definitions.

Practise this in the app: instant marking and a fix for every mistake. Free, no email needed to start.

Practise this lesson

The definitions

sinh x = (e^x - e^-x)/2
cosh x = (e^x + e^-x)/2
tanh x = sinh x / cosh x = (e^2x - 1)/(e^2x + 1)

The reciprocals are sech x = 1/cosh x, cosech x = 1/sinh x (x ≠ 0) and coth x = 1/tanh x (x ≠ 0). Say them as 'shine', 'cosh', 'than', 'shek', 'coshek' and 'coth'.

Graphs and properties

y = cosh x: a U shape through (0, 1), symmetric in the y-axis (an even function). Range cosh x ≥ 1.
y = sinh x: through the origin, rotational symmetry about O (an odd function), range all real numbers.
y = tanh x: through the origin, odd, with horizontal asymptotes y = 1 and y = -1, so -1 < tanh x < 1.
y = sech x: a hump with maximum (0, 1), 0 < sech x ≤ 1.

For large x, cosh x and sinh x are both close to e^x/2.

Function	Domain	Range	Odd or even
sinh x	all x	all real values	odd
cosh x	all x	cosh x ≥ 1	even
tanh x	all x	-1 < tanh x < 1	odd
sech x	all x	0 < sech x ≤ 1	even
cosech x	x ≠ 0	cosech x ≠ 0	odd
coth x	x ≠ 0	\|coth x\| > 1	odd

Exact values from logs

Because e^{ln k} = k and e^{-ln k} = 1/k, values at x = ln k are exact fractions:

cosh(ln k) = (k + 1/k)/2 and sinh(ln k) = (k - 1/k)/2. These come up constantly in exam answers.

Worked example

Find the exact value of sinh(ln 4).

sinh(ln 4) = (e^{ln 4} - e^{-ln 4})/2
= (4 - 1/4)/2
= (15/4)/2

Answer: 15/8

Key idea: sinh x = (e^x - e^-x)/2, cosh x = (e^x + e^-x)/2, tanh x = sinh x/cosh x. cosh x ≥ 1 and is even; sinh and tanh are odd; -1 < tanh x < 1.

Check yourself

Find the exact value of cosh(ln 3). Give your answer as a fraction in its simplest form.
Show answer
5/3. cosh(ln 3) = (3 + 1/3)/2 = (10/3)/2 = 5/3.
Find the exact value of tanh(ln 2). Give your answer as a fraction in its simplest form.
Show answer
3/5. sinh(ln 2) = (2 - 1/2)/2 = 3/4 and cosh(ln 2) = (2 + 1/2)/2 = 5/4, so tanh(ln 2) = 3/5.
Use the definition to evaluate sinh 1.2, to 4 decimal places.
Show answer
1.5095. (e¹.2 - e^-1.2)/2 = (3.3201 - 0.3012)/2 = 1.5095.
What is the range of y = cosh x?
- y ≥ 0
- y ≥ 1
- -1 < y < 1
- all real y
Show answer
y ≥ 1. The minimum of cosh x is cosh 0 = 1, and it grows without limit either side.
Which of these is an odd function?
- cosh x
- sech x
- cosh² x
- sinh x
Show answer
sinh x. sinh(-x) = (e^-x - e^x)/2 = -sinh x.
What are the horizontal asymptotes of y = tanh x?
- y = 0 only
- y = 1 and y = -1
- y = 1 only
- There are none
Show answer
y = 1 and y = -1. tanh x = (e^2x - 1)/(e^2x + 1). As x grows, this tends to 1; as x becomes very negative, e^2x tends to 0 and tanh x tends to -1.
For which values of x is coth x defined?
- x > 0 only
- -1 < x < 1
- x ≥ 1
- x ≠ 0
Show answer
x ≠ 0. coth x = cosh x / sinh x, and sinh x = 0 only when x = 0.

Practise with instant feedback

Sources

Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.