Definition of \tanh x?

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

Is \sinh x odd or even? What is the symmetry of its graph?

Odd; rotational symmetry of order 2 about the origin.

Is \cosh x odd or even? What is the symmetry of its graph?

Even; symmetrical about the y-axis.

What are the asymptotes of y=\tanh x?

The horizontal lines y=1 and y=-1.

Behaviour of \sinh x and \cosh x for large positive x?

Both \approx\frac12e^x, with the gap e^{-x}\to0.

How do you solve an equation like a\cosh x+b\sinh x=c?

Write in exponentials, put u=e^x, solve the quadratic and reject u\le0.

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

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Definition of $\sinh x$?

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Definition of $\sinh x$ ?: $\sinh x=\frac12(e^x-e^{-x})$
Definition of $\cosh x$ ?: $\cosh x=\frac12(e^x+e^{-x})$
Definition of $\tanh x$ ?: $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
Is $\sinh x$ odd or even? What is the symmetry of its graph?: Odd; rotational symmetry of order 2 about the origin.
Is $\cosh x$ odd or even? What is the symmetry of its graph?: Even; symmetrical about the $y$ -axis.
Where is the minimum of $y=\cosh x$ ?: At $(0,1)$ .
What are the asymptotes of $y=\tanh x$ ?: The horizontal lines $y=1$ and $y=-1$ .
Values of $\sinh0$ , $\cosh0$ , $\tanh0$ ?: $0$ , $1$ and $0$ .
Value of $\cosh x-\sinh x$ ?: $e^{-x}$
Behaviour of $\sinh x$ and $\cosh x$ for large positive $x$ ?: Both $\approx\frac12e^x$ , with the gap $e^{-x}\to0$ .
Exact $\sinh(\ln2)$ and $\cosh(\ln2)$ ?: $\frac34$ and $\frac54$
How do you solve an equation like $a\cosh x+b\sinh x=c$ ?: Write in exponentials, put $u=e^x$ , solve the quadratic and reject $u\le0$ .

Exam questions on Hyperbolic functions and their graphs

Let $x=\ln3$ .
Find the exact value of $\tanh x$ .2 marks
Consider the graphs of $y=\cosh x$ and $y=\tanh x$ for all real $x$ .
Describe the symmetry of each of the two graphs.2 marks
The function $f$ is defined for all real $x$ by $f(x)=2\cosh x+3\sinh x$ .
Express $f(x)$ in the form $Ae^x+Be^{-x}$ , where $A$ and $B$ are constants.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).