\tanh x in terms of \sinh x and \cosh x, and in exponentials?

\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}

Define \operatorname{sech}x, \operatorname{cosech}x, \coth x.

\frac{1}{\cosh x}, \frac{1}{\sinh x}, \frac{1}{\tanh x}

Range of \tanh x and its asymptotes?

-1<y<1; asymptotes y=1 and y=-1.

Which of \sinh, \cosh, \tanh are odd?

\sinh x and \tanh x are odd; \cosh x is even.

First step for a\cosh x+b\sinh x=c?

Write in exponentials, multiply by e^x and solve the quadratic in e^x.

Why must you check roots of a quadratic in e^x?

e^x0, so only positive roots give a solution.

\frac12\left(3+\frac13\right)=\frac53

Hyperbolic functions and identitiesEdexcel International A Level Further Maths: Flashcards

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$\sinh x$ in terms of exponentials?

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$\sinh x$ in terms of exponentials?: $\frac{e^x-e^{-x}}{2}$
$\cosh x$ in terms of exponentials?: $\frac{e^x+e^{-x}}{2}$
$\tanh x$ in terms of $\sinh x$ and $\cosh x$ , and in exponentials?: $\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}$
Define $\operatorname{sech}x$ , $\operatorname{cosech}x$ , $\coth x$ .: $\frac{1}{\cosh x}$ , $\frac{1}{\sinh x}$ , $\frac{1}{\tanh x}$
Range of $\cosh x$ ?: $y\ge1$ , with minimum $1$ at $x=0$ .
Range of $\tanh x$ and its asymptotes?: $-1<y<1$ ; asymptotes $y=1$ and $y=-1$ .
Which of $\sinh$ , $\cosh$ , $\tanh$ are odd?: $\sinh x$ and $\tanh x$ are odd; $\cosh x$ is even.
Key identity linking $\cosh^2x$ and $\sinh^2x$ ?: $\cosh^2x-\sinh^2x=1$
$\cosh2x$ in terms of $\cosh^2x$ and $\sinh^2x$ ?: $\cosh2x=\cosh^2x+\sinh^2x$
$\cosh2x$ in terms of $\sinh x$ only?: $\cosh2x=1+2\sinh^2x$
$1-\tanh^2x$ ?: $\operatorname{sech}^2x$
First step for $a\cosh x+b\sinh x=c$ ?: Write in exponentials, multiply by $e^x$ and solve the quadratic in $e^x$ .
Why must you check roots of a quadratic in $e^x$ ?: $e^x>0$ , so only positive roots give a solution.
$\cosh(\ln3)$ ?: $\frac12\left(3+\frac13\right)=\frac53$

Exam questions on Hyperbolic functions and identities

A student evaluates hyperbolic functions at $x=\ln3$ , using $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$ .
Use the identity $\cosh2x=\cosh^2x+\sinh^2x$ to find the exact value of $\cosh(2\ln3)$ .2 marks
The function $f$ is defined by $f(x)=\tanh x$ for all real $x$ .
Show that $\tanh x=\frac{e^{2x}-1}{e^{2x}+1}$ .2 marks
For real $x$ , $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$ .
Prove that $\cosh^2x-\sinh^2x=1$ .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).