Why are they called 'hyperbolic' functions?

They are called hyperbolic because they parametrise the unit hyperbola $x^2 - y^2 = 1$ with the point $(\cosh(t), \sinh(t))$, in the same way that trigonometric functions parametrise the unit circle $x^2 + y^2 = 1$ with the point $(\cos(t), \sin(t))$.

What is the difference between the graph of y = cosh(x) and a parabola like y = x² + 1?

Although they look similar near the origin, they are different curves. The catenary, $y = \cosh(x)$, grows exponentially, so it becomes much steeper than the parabola $y = x^2 + 1$ as $x$ moves away from zero. A parabola is a quadratic function, while a catenary is based on exponential functions.

Do I need to memorise the proofs of the identities?

Yes, it is highly recommended. Exam questions can explicitly ask you to prove an identity from the definitions, as shown in the worked example. Understanding the proofs also deepens your grasp of the relationship between hyperbolic and exponential functions.

When solving an equation, how do I choose between using definitions and using identities?

If the equation contains a mix of different hyperbolic functions (e.g., $\sinh(x)$ and $\cosh(x)$), using the exponential definitions is often the most direct route. If the equation involves powers of a single function (e.g., $\cosh^2(x)$) and another function, try using an identity like $\sinh^2(x) = \cosh^2(x) - 1$ to create a polynomial in one hyperbolic function.

Where can I learn Hyperbolic functions for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 2.1 (Hyperbolic functions) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/2-1-hyperbolic-functions.

Is Hyperbolic functions (2.1) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Hyperbolic functions.

Hyperbolic functions 9231 — Free A-Level…

Hyperbolic functions

Welcome to the world of hyperbolic functions. While their names sound like their trigonometric cousins (sine, cosine, etc.), they are fundamentally defined using the exponential function. They appear in a variety of physical situations, from the shape of a hanging cable to the study of special relativity. In this lesson, we'll build them from the ground up, explore their properties, and learn how to solve equations with them.

BY THE END, YOU CAN…

1Define the six hyperbolic functions in terms of the exponential function.
2Understand, prove, and use fundamental hyperbolic identities.
3Sketch the graphs of sinh(x), cosh(x), and tanh(x), stating their key features.
4Solve equations involving hyperbolic functions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Exponential Cousins of Trigonometry

Hyperbolic functions are special combinations of the exponential function e^x. They behave in ways very similar to trigonometric functions, but are based on a hyperbola instead of a circle.

THINK OF IT LIKE…

Imagine a heavy chain or rope hanging between two posts of equal height. The curve it forms is not a parabola, but a 'catenary', which is precisely the graph of the hyperbolic cosine function, y = cosh(x). Just as sine and cosine describe points on a circle, hyperbolic sine and cosine describe points on a hyperbola.

1
Learn the definitions of sinh(x) and cosh(x) using e^x and e^-x.
2
Derive the definitions for tanh(x) and the reciprocal functions (sech, cosech, coth).
3
Master the core identity cosh²(x) - sinh²(x) = 1 and its variations.
4
Sketch the graphs, paying close attention to key points, symmetry, and asymptotes.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$sinh(x) = \frac{e^x - e^{-x}}{2} \ cosh(x) = \frac{e^x + e^{-x}}{2}$

Tap a symbol — great for exam definitions

$tanh(x) = \frac{\sinh(x)}{\cosh(x)} \ sech(x) = \frac{1}{\cosh(x)} \ cosech(x) = \frac{1}{\sinh(x)} \ coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

1. Definitions of Hyperbolic Functions

The two primary hyperbolic functions are the hyperbolic sine (sinh, pronounced 'shine') and the hyperbolic cosine (cosh). They are defined as specific combinations of the exponential functions $e^x$ and $e^{-x}$ .

$sinh(x) = \frac{e^x - e^{-x}}{2} \\ cosh(x) = \frac{e^x + e^{-x}}{2}$

From these two, we can define the other four hyperbolic functions in a way that directly mirrors trigonometry. The hyperbolic tangent (tanh) is sinh divided by cosh, and the reciprocal functions are sech (hyperbolic secant), cosech (hyperbolic cosecant), and coth (hyperbolic cotangent).

$tanh(x) = \frac{\sinh(x)}{\cosh(x)} \\ sech(x) = \frac{1}{\cosh(x)} \\ cosech(x) = \frac{1}{\sinh(x)} \\ coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)}$

2. Fundamental Hyperbolic Identities

Just as trigonometric functions are linked by identities like $\sin^2(\theta) + \cos^2(\theta) = 1$ , hyperbolic functions have their own set of identities. These can all be proven by substituting the exponential definitions. The most important one is the hyperbolic equivalent of the Pythagorean identity.

$\cosh^2(x) - \sinh^2(x) = 1$
$1 - \tanh^2(x) = \text{sech}^2(x)$ (Divide the first identity by $\cosh^2(x)$ )
$\coth^2(x) - 1 = \text{cosech}^2(x)$ (Divide the first identity by $\sinh^2(x)$ )

3. Graphs of Hyperbolic Functions

The graphs of hyperbolic functions have distinct shapes that you must be able to recognise and sketch. They can be understood by considering them as the sum or ratio of exponential graphs.

y = sinh(x): An odd function, passing through the origin (0,0). It is strictly increasing. Domain: $x \in \mathbb{R}$ . Range: $y \in \mathbb{R}$ .
y = cosh(x): An even function, symmetric about the y-axis. It has a minimum value of 1 at the point (0,1). The shape is a catenary. Domain: $x \in \mathbb{R}$ . Range: $y \ge 1$ .
y = tanh(x): An odd function, passing through the origin (0,0). It has two horizontal asymptotes at $y=1$ (as $x \to \infty$ ) and $y=-1$ (as $x \to -\infty$ ). Domain: $x \in \mathbb{R}$ . Range: $-1 < y < 1$ .

EXAM TIP

Be very careful with the identities. The minus sign in $\cosh^2(x) - \sinh^2(x) = 1$ is a common source of errors. Always double-check if you are using a hyperbolic or a trigonometric identity. When sketching $y=\cosh(x)$ , ensure your minimum is at (0,1) and the curve is U-shaped, but distinct from a parabola (it's steeper).

4. Solving Hyperbolic Equations

Equations involving hyperbolic functions can typically be solved using one of two main strategies. The first is to use the exponential definitions to transform the equation into one involving $e^x$ , which often leads to a hidden quadratic. The second is to use hyperbolic identities to simplify the equation into a polynomial in terms of one hyperbolic function, such as a quadratic in $\cosh(x)$ .

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Prove the identity $\cosh^2(x) - \sinh^2(x) = 1$ from the exponential definitions of $\sinh(x)$ and $\cosh(x)$ .

1
We start with the left-hand side (LHS) and substitute the definitions.

EXAMPLE 2

Solve the equation $3\cosh(x) + 2\sinh(x) = 4$ , giving your answer in an exact logarithmic form.

1
We will use the exponential definitions for $\cosh(x)$ and $\sinh(x)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAHyperbolic functions

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What is the definition of sinh(x)?

TAP OR SPACE TO FLIP

ANSWER

sinh(x) = (e^x - e^-x) / 2

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
$\cosh^2(x) - \sinh^2(x) = 1$
✓
$1 - \tanh^2(x) = \text{sech}^2(x)$ (Divide the first identity by $\cosh^2(x)$ )
✓
$\coth^2(x) - 1 = \text{cosech}^2(x)$ (Divide the first identity by $\sinh^2(x)$ )

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Hyperbolic Functions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Hyperbolic Functions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.7 Proof by induction NEXT →2.2 Matrices

Hyperbolic functions

BY THE END, YOU CAN…

1Define the six hyperbolic functions in terms of the exponential function.
2Understand, prove, and use fundamental hyperbolic identities.
3Sketch the graphs of sinh(x), cosh(x), and tanh(x), stating their key features.
4Solve equations involving hyperbolic functions.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Exponential Cousins of Trigonometry

Hyperbolic functions are special combinations of the exponential function e^x. They behave in ways very similar to trigonometric functions, but are based on a hyperbola instead of a circle.

THINK OF IT LIKE…

1
Learn the definitions of sinh(x) and cosh(x) using e^x and e^-x.
2
Derive the definitions for tanh(x) and the reciprocal functions (sech, cosech, coth).
3
Master the core identity cosh²(x) - sinh²(x) = 1 and its variations.
4
Sketch the graphs, paying close attention to key points, symmetry, and asymptotes.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$sinh(x) = \frac{e^x - e^{-x}}{2} \ cosh(x) = \frac{e^x + e^{-x}}{2}$

Tap a symbol — great for exam definitions

$tanh(x) = \frac{\sinh(x)}{\cosh(x)} \ sech(x) = \frac{1}{\cosh(x)} \ cosech(x) = \frac{1}{\sinh(x)} \ coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)}$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

1. Definitions of Hyperbolic Functions

$sinh(x) = \frac{e^x - e^{-x}}{2} \\ cosh(x) = \frac{e^x + e^{-x}}{2}$

$tanh(x) = \frac{\sinh(x)}{\cosh(x)} \\ sech(x) = \frac{1}{\cosh(x)} \\ cosech(x) = \frac{1}{\sinh(x)} \\ coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)}$

2. Fundamental Hyperbolic Identities

$\cosh^2(x) - \sinh^2(x) = 1$
$1 - \tanh^2(x) = \text{sech}^2(x)$ (Divide the first identity by $\cosh^2(x)$ )
$\coth^2(x) - 1 = \text{cosech}^2(x)$ (Divide the first identity by $\sinh^2(x)$ )

3. Graphs of Hyperbolic Functions

The graphs of hyperbolic functions have distinct shapes that you must be able to recognise and sketch. They can be understood by considering them as the sum or ratio of exponential graphs.

y = sinh(x): An odd function, passing through the origin (0,0). It is strictly increasing. Domain: $x \in \mathbb{R}$ . Range: $y \in \mathbb{R}$ .
y = cosh(x): An even function, symmetric about the y-axis. It has a minimum value of 1 at the point (0,1). The shape is a catenary. Domain: $x \in \mathbb{R}$ . Range: $y \ge 1$ .
y = tanh(x): An odd function, passing through the origin (0,0). It has two horizontal asymptotes at $y=1$ (as $x \to \infty$ ) and $y=-1$ (as $x \to -\infty$ ). Domain: $x \in \mathbb{R}$ . Range: $-1 < y < 1$ .

EXAM TIP

4. Solving Hyperbolic Equations

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

Prove the identity $\cosh^2(x) - \sinh^2(x) = 1$ from the exponential definitions of $\sinh(x)$ and $\cosh(x)$ .

1
We start with the left-hand side (LHS) and substitute the definitions.

EXAMPLE 2

Solve the equation $3\cosh(x) + 2\sinh(x) = 4$ , giving your answer in an exact logarithmic form.

1
We will use the exponential definitions for $\cosh(x)$ and $\sinh(x)$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAHyperbolic functions

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

What is the definition of sinh(x)?

TAP OR SPACE TO FLIP

ANSWER

sinh(x) = (e^x - e^-x) / 2

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
$\cosh^2(x) - \sinh^2(x) = 1$
✓
$1 - \tanh^2(x) = \text{sech}^2(x)$ (Divide the first identity by $\cosh^2(x)$ )
✓
$\coth^2(x) - 1 = \text{cosech}^2(x)$ (Divide the first identity by $\sinh^2(x)$ )

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Hyperbolic Functions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Hyperbolic Functions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.7 Proof by induction NEXT →2.2 Matrices

Revise Hyperbolic functions for Cambridge 9231 (2.1)

In simple terms

Explore the concept

Key formulas

Full topic notes

1. Definitions of Hyperbolic Functions

2. Fundamental Hyperbolic Identities

3. Graphs of Hyperbolic Functions

4. Solving Hyperbolic Equations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Hyperbolic Functions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

1. Definitions of Hyperbolic Functions

2. Fundamental Hyperbolic Identities

3. Graphs of Hyperbolic Functions

4. Solving Hyperbolic Equations

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Hyperbolic Functions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Hyperbolic functions