# Further Maths - Hyperbolic Functions

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/hyperbolic-functions/ · Updated: 2021-03-15 · Tags: further-maths, school

## See Also
## Flashcards
### 2021-03-15
##### $$\sinh x$$ What is the definition??
$$
\frac{e^x - e^{-x}}{2}
$$

##### $$\cosh x$$ What is the definition??
$$
\frac{e^x + e^{-x}}{2}
$$

##### $$\tanh x$$ What is the definition??
$$
\frac{e^{2x} - 1}{e^{2x} + 1}
$$

##### ![PHOTO SINH GRAPH](sinh-graph.gif) What function is this??
$$
\sinh
$$

##### ![PHOTO COSH GRAPH](cosh-graph.gif) What function is this??
$$
\cosh
$$

##### ![PHOTO TANH GRAPH](tanh-graph.gif) What function is this??
$$
\tanh
$$

##### $$y = \sinh x$$ What does the graph look like??
![PHOTO SINH GRAPH](sinh-graph.gif)

##### $$y = \cosh x$$ What does the graph look like??
![PHOTO COSH GRAPH](cosh-graph.gif)

##### $$y = \tanh x$$ What does the graph look like??
![PHOTO TANH GRAPH](tanh-graph.gif)

##### What is true about any value of $\cosh x$??
It is above $1$.

##### $$e^x - e^{-x} = 10$$ How would you rewrite this??
$$
e^2x - 1 = 10e^x
$$

### 2021-03-16
##### $$\arcsinh x$$ What is the definition??
$$
\ln(x + \sqrt{x^2 + 1})
$$

##### $$\arcosh x$$ What is the definition??
$$
\ln(x + \sqrt{x^2 - 1})
$$

##### $$\artanh x$$ What is the definition??
$$
\frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)
$$

##### What is the domain for $\arcosh x$??
$$
x \ge 1
$$

#### What is the domain for $\artanh x$??
$$
|x| < 1
$$
##### ![PHOTO ARSINH GRAPH](arsinh-graph.gif) What function is this??
$$
\arsinh
$$

##### ![PHOTO ARCOSH GRAPH](arcosh-graph.gif) What function is this??
$$
\arcosh
$$

##### ![PHOTO ARTANH GRAPH](artanh-graph.gif) What function is this??
$$
\artanh
$$

##### $$y = \arsinh x$$ What does the graph look like??
![PHOTO ARSINH GRAPH](arsinh-graph.gif)

##### $$y = \arcosh x$$ What does the graph look like??
![PHOTO ARCOSH GRAPH](arcosh-graph.gif)

##### $$y = \artanh x$$ What does the graph look like??
![PHOTO ARTANH GRAPH](artanh-graph.gif)

##### What is true about any value of $\cosh x$??
It is above $1$.

### 2021-03-17
##### What is Osborn's Rule??
Replace any product of two $\sin$ terms by minus the products of two $\sin$ terms.

##### By Osborn's Rule, what is $\sinA\sinB$ in hyperbolic functions??
$$
-\sinhA\sinhB
$$

##### By Osborn's Rule, what is $\tan^2 x$ in hyperbolic functions??
$$
-\tanh^2 x
$$

##### How do you convert a trig identity to a hyperbolic trig identity??
* Replace all normal functions with their hyperbolic equivalents
* Use Osborn's Rule

##### If you're not allowed to use Osborn's Rule when converting a hyperbolic trig identity, what can you do??
Use the $e^x$ defintitions of all the functions.

##### $$\sin^2 x + \cos^2 x = 1$$ What is the hyperbolic equivalent??
$$
\cos^2 x - \sin^2 x = 1
$$

##### $$\frac{d}{dx} \sinh x$$ What is this equal to??
$$
\cosh x
$$

##### $$\frac{d}{dx} \cosh x$$ What is this equal to??
$$
\sinh x
$$

##### $$\frac{d}{dx} \tanh x$$ What is this equal to??
$$
\sech^2 x
$$

##### $$\frac{d}{dx} (\sinh^{-1} x)$$ What is the equal to??
$$
\frac{1}{\frac{x^2 + 1}}
$$

##### $$\frac{d}{dx} (\cosh^{-1} x)$$ What is the equal to??
$$
\frac{1}{\frac{x^2 - 1}}
$$

##### $$\frac{d}{dx} (\tanh^{-1} x)$$ What is the equal to??
$$
\frac{1}{\frac{1 - x^2}}
$$

### 2021-03-24
##### If $y = \sinh^{-1}(x)$, what is $x$ equal to??
$$
x = \sinh(y)
$$

##### $$x = \sinh(y)$$ What do you get if you differentiate both sides??
$$
\frac{dx}{dy} = \cosh(y)
$$

##### $$\frac{dx}{dy} = \cosh(y)$$ The aim here is to get $\frac{dy}{dx}$. How could you write $\cosh(y)$ made out of something you already know??
$$
\frac{dx}{dy} = \sqrt{1 + \sinh^2(y)}
$$

##### $$\frac{dx}{dx} = \sqrt{1 + \sinh^2(x)}$$ How could you rewrite this in terms of what you already know??
$$
\frac{dx}{dy} = \sqrt{1 + x^2]}
$$

##### $$\frac{dx}{dy} = u$$ How could you rewrite this so it's $\frac{dy}{dx}$??
$$
\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}
$$

##### When finding the derivative of an inverse function, what's the trick??
Rewriting some $f(y)$ in terms of $x$.

### 2021-03-25
##### $$\int \frac{1}{\sqrt{x^2 + 1}}dx$$ What is this equal to??
$$
\sinh^{-1} x
$$

##### $$\int \frac{1}{\sqrt{x^2 - 1}}dx$$ What is this equal to??
$$
\cosh^{-1} x
$$

##### $$\frac{d}{dx}\left(\sinh^{-1}\left(\frac{x}{a}\right)\right)$$ What is this equal to??
$$
\frac{1}{\sqrt{x^2 + a^2}}
$$

##### $$\frac{d}{dx}\left(\cosh^{-1}\left(\frac{x}{a}\right)\right)$$ What is this equal to??
$$
\frac{1}{\sqrt{x^2 - a^2}}
$$

##### $$\int\frac{1}{\sqrt{x^2 + a^2}}dx$$ What is this equal to??
$$
\sinh^{-1}\left(\frac{x}{a}\right) \pmb{+ c}
$$

##### $$\int\frac{1}{\sqrt{x^2 - a^2}}dx$$ What is this equal to??
$$
\cosh^{-1}\left(\frac{x}{a}\right) \pmb{+ c}
$$

##### $$\int\frac{1}{\sqrt{x^2 - 16}}dx$$ What is this equal to??
$$
\cosh^{-1}\left(\frac{x}{4}\right) \pmb{+ c}
$$

##### $$\int\frac{1}{\sqrt{x^2 + 8}}dx$$ What is this equal to??
$$
\sinh^{-1}\left(\frac{x}{2\sqrt{2}}\right) \pmb{+ c}
$$

##### $$\sqrt{4x^2 + 1}$$ How could you rewrite this to aid with integrating??
$$
2\sqrt{x^2 + \frac{1}{2}}
$$

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