# Hyperbolic Functions

> CIE A-Level Further Mathematics · Further Pure 2
> Source: https://www.owlsprep.com/study/cie-9231-u2-hyperbolic-functions/

This module covers definitions, graphs, identities, inverse functions, and calculus operations for hyperbolic functions, aligned to CIE 9231 A-Level Further Mathematics exam requirements.

**Prerequisites:** [Exponential and logarithmic functions](https://www.owlsprep.com/study/cie-9231-p1-exponential-logarithms/); [Basic differentiation and integration](https://www.owlsprep.com/study/cie-9231-p1-differentiation-integration/)

## Learning objectives

- Define hyperbolic functions in terms of exponential functions
- Sketch graphs and state key properties of hyperbolic and inverse hyperbolic functions
- Prove and apply standard hyperbolic identities
- Differentiate and integrate hyperbolic and inverse hyperbolic functions
- Solve equations involving hyperbolic functions

## Definitions and Graphs of Hyperbolic Functions

**Core Hyperbolic Functions** — All hyperbolic functions are defined in terms of exponentials: $\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$, $\tanh x = \frac{\sinh x}{\cosh x}$, $\coth x = \frac{1}{\tanh x}$, $\text{sech}\,x = \frac{1}{\cosh x}$, $\text{csch}\,x = \frac{1}{\sinh x}$

*Notation:* $\sinh x$, $\cosh x$, $\tanh x$, $\coth x$, $\text{sech}\,x$, $\text{csch}\,x$

*Example:* $\sinh 0 = 0$, $\cosh 0 = 1$, $\tanh 0 = 0$

Hyperbolic functions share many similarities with trigonometric functions, but have different parity and range properties:

- $\cosh x$ is even ($\cosh(-x) = \cosh x$), range $[1, \infty)$
- $\sinh x$ is odd ($\sinh(-x) = -\sinh x$), range $(-\infty, \infty)$
- $\tanh x$ is odd, range $(-1, 1)$

**Worked example:** Find the exact value of $\cosh(\ln 3)$

1. Start with the definition of $\cosh x$:

   $$\cosh x = \frac{e^x + e^{-x}}{2}$$
2. Substitute $x = \ln 3$, using the property $e^{k\ln a} = a^k$:

   $$e^{\ln 3} = 3, \quad e^{-\ln 3} = \frac{1}{3}$$
3. Calculate and simplify:

   $$\cosh(\ln 3) = \frac{3 + \frac{1}{3}}{2} = \frac{10}{6} = \frac{5}{3}$$

## Hyperbolic Identities

> **Osborn's Rule**
>
> To convert a trigonometric identity to a hyperbolic identity: replace $\cos \to \cosh$, $\sin \to \sinh$, and flip the sign of any term that contains a product of two sines.

**Fundamental Pythagorean Hyperbolic Identity** — The core identity for all hyperbolic functions is: $\cosh^2 x - \sinh^2 x = 1$

- $\sinh(A+B) = \sinh A \cosh B + \cosh A \sinh B$
- $\cosh(A+B) = \cosh A \cosh B + \sinh A \sinh B$
- $\cosh 2x = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1$
- $\sinh 2x = 2\sinh x \cosh x$

**Worked example:** Prove that $\cosh^2 x - \sinh^2 x = 1$ using exponential definitions

1. Substitute the definitions of $\cosh x$ and $\sinh x$ into the left-hand side:

   $$\cosh^2 x - \sinh^2 x = \left(\frac{e^x + e^{-x}}{2}\right)^2 - \left(\frac{e^x - e^{-x}}{2}\right)^2$$
2. Expand both squares:

   $$= \frac{e^{2x} + 2 + e^{-2x}}{4} - \frac{e^{2x} - 2 + e^{-2x}}{4}$$
3. Simplify the numerator:

   $$= \frac{(e^{2x} + 2 + e^{-2x}) - (e^{2x} - 2 + e^{-2x})}{4} = \frac{4}{4} = 1$$
4. This matches the right-hand side, completing the proof.

## Inverse Hyperbolic Functions

Restricted domains make core hyperbolic functions one-to-one, so we can define inverse functions that can be written exactly in terms of natural logarithms.

**Logarithmic Forms of Inverse Hyperbolic Functions** — $\text{arsinh}\,x = \ln\left(x + \sqrt{x^2 + 1}\right)$ for all $x \in \mathbb{R}$; $\text{arcosh}\,x = \ln\left(x + \sqrt{x^2 - 1}\right)$ for $x \geq 1$; $\text{artanh}\,x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ for $|x| < 1$

*Notation:* $\text{arsinh}\,x$, $\text{arcosh}\,x$, $\text{artanh}\,x$

**Worked example:** Express $\text{arsinh}\,3$ as an exact natural logarithm

1. Use the standard logarithmic form for $\text{arsinh}\,x$:

   $$\text{arsinh}\,x = \ln\left(x + \sqrt{x^2 + 1}\right)$$
2. Substitute $x = 3$:

   $$x^2 + 1 = 9 + 1 = 10$$
3. Final result:

   $$\text{arsinh}\,3 = \ln\left(3 + \sqrt{10}\right)$$

**Check your understanding**

Test your understanding of domain restrictions

1. What is the domain of $\text{arcosh}\,x$?

   - All real numbers
   - $x \geq 1$
   - $|x| < 1$
   - $x > 0$

   *Answer:* $x \geq 1$

   *Why:* Correct: $\cosh x$ has range $[1, \infty)$, so the inverse function $\text{arcosh}\,x$ has domain $[1, \infty)$

## Calculus of Hyperbolic Functions

Differentiating and integrating hyperbolic and inverse hyperbolic functions gives standard results that are frequently used for integrals involving quadratics under square roots.

| Function | Derivative |
| --- | --- |
| $\sinh x$ | $\cosh x$ |
| $\cosh x$ | $\sinh x$ |
| $\tanh x$ | $\text{sech}^2 x$ |
| $\text{arsinh}\,x$ | $\frac{1}{\sqrt{x^2+1}}$ |
| $\text{arcosh}\,x$ | $\frac{1}{\sqrt{x^2-1}}$ |
| $\text{artanh}\,x$ | $\frac{1}{1-x^2}$ |

**Worked example:** Find $\int \frac{1}{\sqrt{9x^2 - 16}} dx$ for $x > \frac{4}{3}$

1. Factor out constants to match the standard integral form for inverse hyperbolic functions:

   $$\int \frac{1}{\sqrt{9x^2 - 16}} dx = \frac{1}{3} \int \frac{1}{\sqrt{(3x)^2 - 4^2}} d(3x)$$
2. Use the standard result $\int \frac{1}{\sqrt{u^2 - a^2}} du = \text{arcosh}\left(\frac{u}{a}\right) + C$:

   $$= \frac{1}{3} \text{arcosh}\left(\frac{3x}{4}\right) + C$$
3. This can also be rewritten in logarithmic form if required:

   $$= \frac{1}{3} \ln\left(3x + \sqrt{9x^2 - 16}\right) + C'$$

> **exam_tip**
>
> Always check the domain of your result, as the form of the integral changes for different ranges of $x$.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Sign error in hyperbolic identities when using Osborn's rule
  - Why it fails: Students often forget to flip the sign for terms containing a product of two sines
  - Correct: Apply Osborn's rule systematically: flip the sign of any term that has a product of two hyperbolic sines
- **Wrong:** Using $\text{arcosh}\,x$ for $x < 1$
  - Why it fails: The function is undefined for $x < 1$, so any result with this domain is invalid
  - Correct: Always check the argument of $\text{arcosh}\,x$ is at least 1 before using the function
- **Wrong:** Writing derivative of $\tanh x$ as $-\text{sech}^2 x$
  - Why it fails: Confusion with derivative of $\tan x$ leads to incorrect negative sign
  - Correct: Recall $\frac{d}{dx}\tanh x = \text{sech}^2 x$, there is no negative sign for this derivative
- **Wrong:** Missing the constant factor for linear substitutions in integrals
  - Why it fails: When substituting $u = kx$, $du = kdx$ so $dx = \frac{du}{k}$, this factor is often missed
  - Correct: Always adjust the integral for the derivative of the substitution to get the correct constant factor in the final result

## Cheatsheet

| Function | Definition | Derivative | Inverse Log Form |
| --- | --- | --- | --- |
| $\sinh x$ | $\frac{e^x - e^{-x}}{2}$ | $\cosh x$ | $\ln(x + \sqrt{x^2+1})$ |
| $\cosh x$ | $\frac{e^x + e^{-x}}{2}$ | $\sinh x$ | $\ln(x + \sqrt{x^2-1}), x\geq 1$ |
| $\tanh x$ | $\frac{\sinh x}{\cosh x}$ | $\text{sech}^2 x$ | $\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right), \|x\|<1$ |
| Key Identity | - | - | $\cosh^2 x - \sinh^2 x = 1$ |
| $\int \frac{1}{\sqrt{x^2+a^2}}dx$ | - | - | $\text{arsinh}\left(\frac{x}{a}\right)+C$ |
| $\int \frac{1}{\sqrt{x^2-a^2}}dx$ | - | - | $\text{arcosh}\left(\frac{x}{a}\right)+C$ |

## What's next

Hyperbolic functions are a foundational topic for Further Pure 2, and they appear regularly in integration problems, arc length calculations, coordinate geometry, and solving differential equations. Mastery of their identities and calculus results simplifies more advanced topics like reduction formulae and improper integrals. Many integration results that use inverse trigonometric functions can also be expressed in terms of inverse hyperbolic functions, so this topic helps you connect different areas of calculus and prepare for mixed questions in the exam.

- [Further Differentiation and Applications](https://www.owlsprep.com/study/cie-9231-u2-further-differentiation-and-applications/)
- [Further integration and applications](https://www.owlsprep.com/study/cie-9231-u2-further-integration-and-applications/)
- [Complex Numbers](https://www.owlsprep.com/study/cie-9231-u2-complex-numbers/)

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