Definition of \tanh x?

\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}

Domain and range of \sinh x?

Domain \mathbb{R}, range \mathbb{R}.

Domain and range of \cosh x?

Domain \mathbb{R}, range \cosh x\ge1.

Domain and range of \tanh x?

Domain \mathbb{R}, range -1<\tanh x<1.

Which hyperbolic function is even?

\cosh x (symmetric about the y-axis).

How do you solve \cosh x=k from the definition?

Write e^x+e^{-x}=2k, multiply by e^x, solve the quadratic in e^x, then take \ln.

Hyperbolic functions and their graphsEdexcel A-Level Further Maths: Flashcards

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Definition of $\sinh x$?

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Definition of $\sinh x$ ?: $\frac12\left(e^x-e^{-x}\right)$
Definition of $\cosh x$ ?: $\frac12\left(e^x+e^{-x}\right)$
Definition of $\tanh x$ ?: $\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
$\tanh x$ in terms of $e^{2x}$ ?: $\frac{e^{2x}-1}{e^{2x}+1}$
What is $\cosh x+\sinh x$ ?: $e^x$
Domain and range of $\sinh x$ ?: Domain $\mathbb{R}$ , range $\mathbb{R}$ .
Domain and range of $\cosh x$ ?: Domain $\mathbb{R}$ , range $\cosh x\ge1$ .
Domain and range of $\tanh x$ ?: Domain $\mathbb{R}$ , range $-1<\tanh x<1$ .
Which hyperbolic function is even?: $\cosh x$ (symmetric about the $y$ -axis).
Which hyperbolic functions are odd?: $\sinh x$ and $\tanh x$ .
Minimum point of $y=\cosh x$ ?: $(0,1)$
Asymptotes of $y=\tanh x$ ?: $y=1$ and $y=-1$ .
Value of $\sinh(\ln2)$ and $\cosh(\ln2)$ ?: $\frac34$ and $\frac54$
How do you solve $\cosh x=k$ from the definition?: Write $e^x+e^{-x}=2k$ , multiply by $e^x$ , solve the quadratic in $e^x$ , then take $\ln$ .

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).