Hyperbolic functions and their graphsEdexcel A-Level Further Maths: Revision notes

Section 1

Definitions

The hyperbolic functions are defined using the exponential function: $\sinh x=\frac12\left(e^x-e^{-x}\right),\quad\cosh x=\frac12\left(e^x+e^{-x}\right),\quad\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}.$ Multiplying top and bottom of $\tanh x$ by $e^x$ gives $\tanh x=\frac{e^{2x}-1}{e^{2x}+1}$ . Adding and subtracting the definitions: $\cosh x+\sinh x=e^x$ and $\cosh x-\sinh x=e^{-x}$ . Example: $\sinh(\ln3)=\frac12\left(3-\frac13\right)=\frac43$ , $\cosh(\ln3)=\frac53$ and $\tanh(\ln3)=\frac45$ .

Key termssinhcoshtanh

Common mistake

Confusing the signs: $\cosh$ has $+$ and $\sinh$ has $-$ between the exponentials.

Exam tip

Remember $e^{\ln a}=a$ and $e^{-\ln a}=\frac1a$ when evaluating at $x=\ln a$ .

Section 2

Domains and ranges

All three functions are defined for every real $x$ (domain $\mathbb{R}$ ).

$\sinh x$ : range all real numbers.
$\cosh x$ : range $\cosh x\ge1$ , because $e^x+e^{-x}\ge2$ , with equality at $x=0$ .
$\tanh x$ : range $-1<\tanh x<1$ . Use the range to explain why equations like $\cosh x=\frac12$ or $\tanh x=\frac32$ have no real solutions.

Key termsdomainrange

Common mistake

Writing $-1\le\tanh x\le1$ . The asymptotes $\pm1$ are never reached.

Section 3

Graphs and symmetry

$y=\sinh x$ : passes through the origin with gradient $1$ , odd ( $\sinh(-x)=-\sinh x$ , rotational symmetry about the origin), increasing everywhere, and like $\frac12e^x$ for large positive $x$ .
$y=\cosh x$ : even ( $\cosh(-x)=\cosh x$ , symmetric about the $y$ -axis), minimum point $(0,1)$ , U-shaped (a catenary), and like $\frac12e^{|x|}$ for large $|x|$ . Always above $y=\sinh x$ .
$y=\tanh x$ : odd, passes through the origin with gradient $1$ , increasing, with horizontal asymptotes $y=1$ and $y=-1$ .

Key termsodd functioneven functionasymptote

Exam tip

When sketching, label the intercepts $(0,0)$ or $(0,1)$ and write the asymptote equations.

Section 4

Evaluating and solving using the definitions

To solve an equation such as $\cosh x=\frac{13}{12}$ , replace $\cosh x$ by $\frac12(e^x+e^{-x})$ and multiply through by $e^x$ to get a quadratic in $e^x$ : $6e^{2x}-13e^x+6=0\ \Rightarrow\ (2e^x-3)(3e^x-2)=0\ \Rightarrow\ e^x=\frac32\text{ or }\frac23.$ Then take natural logarithms: $x=\pm\ln\frac32$ . Two solutions for $\cosh$ (even), but only one for $\sinh x=k$ (increasing). Always check each root against the range: $e^x$ must be positive.

Key termsquadratic in $e^x$

Common mistake

Taking $\ln$ of a negative value of $e^x$ . Reject it.

Section 5

Modelling with hyperbolic functions

A hanging cable or chain takes the shape of $y=a\cosh\frac xa$ (a catenary), with its lowest point $(0,a)$ . Because $\cosh$ is even the cable is symmetric about the $y$ -axis, so poles at equal distances either side have equal heights. For $y=4\cosh\frac x4$ , a pole at $x=8$ has height $4\cosh2=2\left(e^2+e^{-2}\right)\approx15.05$ m. Questions ask you to find heights or positions by substituting into the definition, then interpret the answer in context, with units.

Key termscatenary

Exam tip

Give the exact form first, then a decimal to the accuracy asked.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Hyperbolic functions and their graphs

Let $x=\ln3$ .
Hence, or otherwise, find the value of $\cosh x+\sinh x$ .2 marks
The functions $f$ , $g$ and $h$ are defined for all real $x$ by $f(x)=\sinh x$ , $g(x)=\cosh x$ and $h(x)=\tanh x$ .
Explain why the equation $g(x)=\frac12$ has no real solutions.2 marks
The equation $\cosh x=\frac{13}{12}$ has two real solutions.
Use the definition of $\cosh x$ to show that $6e^{2x}-13e^x+6=0$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).