# Further Maths - Hyperbolic Functions

## See Also

## Flashcards

### 2021-03-15

## \[\sinh x\]
What is the definition?

## \[\cosh x\]
What is the definition?

## \[\tanh x\]
What is the definition?

What function is this?

What function is this?

What function is this?

## \[y = \sinh x\]
What does the graph look like?

## \[y = \cosh x\]
What does the graph look like?

## \[y = \tanh x\]
What does the graph look like?

What is true about any value of $\cosh x$?

It is above $1$.

## \[e^x - e^{-x} = 10\]
How would you rewrite this?

### 2021-03-16

## \[\arcsinh x\]
What is the definition?

## \[\arcosh x\]
What is the definition?

## \[\artanh x\]
What is the definition?

What is the domain for $\arcosh x$?

#### What is the domain for $\artanh x$??

\[|x| < 1\] What function is this?

What function is this?

What function is this?

## \[y = \arsinh x\]
What does the graph look like?

## \[y = \arcosh x\]
What does the graph look like?

## \[y = \artanh x\]
What does the graph look like?

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?

By Osborn’s Rule, what is $\tan^2 x$ in hyperbolic functions?

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?

## \[\frac{d}{dx} \sinh x\]
What is this equal to?

## \[\frac{d}{dx} \cosh x\]
What is this equal to?

## \[\frac{d}{dx} \tanh x\]
What is this equal to?

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

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

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

### 2021-03-24

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

## \[x = \sinh(y)\]
What do you get if you differentiate both sides?

## \[\frac{dx}{dy} = \cosh(y)\]
The aim here is to get $\frac{dy}{dx}$. How could you write $\cosh(y)$ made out of something you already know?

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

## \[\frac{dx}{dy} = u\]
How could you rewrite this so it’s $\frac{dy}{dx}$?

When finding the derivative of an inverse function, what’s the trick?

Rewriting some $f(y)$ in terms of $x$.