Hyperbolic Trigonometric Functions.md

Definitions

Hyperbolic functions are functions with similar properties to [[Trigonometric Functions]] but are defined in terms of exponentials: $$ \sinh(x) \equiv \frac{e^x - e^{-x}}{2} \quad \quad \cosh(x) \equiv \frac{e^x + e^{-x}}{2}\quad \quad \tanh(x) \equiv \frac{\sinh(x)}{\cosh(x)} = \frac{e^{2x} - 1}{e^{2x} + 1} $$ $$ \begin{aligned} \text{cosech}(x) \equiv &\frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}} \quad \quad \text{sech}(x) \equiv \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \ &\coth(x) \equiv \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} = \frac{e^{2x} + 1}{e^{2x} - 1} \end{aligned} $$

---

A-Level Further Core 2

Properties

Parity

$\sinh(x)$ is odd, so $\sinh(-x) = -\sinh(x), a \in \mathbb{R}$ $\cosh(x)$ is even, so $\cosh(-x) = \cosh(x), a \in \mathbb{R}$ $\tanh(x)$ is odd, so $\tanh(-x) = -\tanh(x), a \in \mathbb{R}$

Range and Domain

$\sinh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}$ $\cosh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}, y \geq 1$ $\tanh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}, -1 < y < 1$

Inverse Hyperbolic Functions

Hyperbolic function	Inverse	Natural Log Form
$y = \sinh(x)$	$y = \text{arsinh}(x)$	$\text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1})$
$y = \cosh(x), x \geq 0$	$y = \text{arcosh}(x), x \geq 1$	$\text{arcosh}(x) = \pm\ln(x + \sqrt{x^2 - 1})$
$y = \tanh(x)$	$y = \text{artanh}(x)$, $\lvert x \rvert$ $< 1$	$\displaystyle\text{artanh}(x) = \frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)$

Identities and Equations

Generally given a trig identity you can find the corresponding hyperbolic identity with Osborn's Rule $$\cos(x) \rightarrow \cosh(x) \quad \sin(x) \rightarrow \sinh(x) \quad \sin^2(x) \rightarrow -\sinh^2(x) \quad\sin A \sin B \rightarrow -\sinh(A)\sinh(B)$$

Calculus With Hyperbolic Functions

Most found in formula booklet

Differentiation

$$\begin{aligned} &\frac{d}{dx} \sinh(x) = \cosh(x) &&\frac{d}{dx} \cosh(x) = \sinh(x) \\ &\frac{d}{dx} \tanh(x) = \text{sech}^2(x) &&\frac{d}{dx} \text{arsinh}(x) = \frac{1}{\sqrt{x^2+1}} \\ &\frac{d}{dx} \text{arcosh}(x) = \frac{1}{\sqrt{x^2-1}} &&\frac{d}{dx} \text{artanh}(x) = \frac{1}{1-x^2} \end{aligned}$$

Integration

$$\begin{aligned} &\int \sinh(x) , dx = \cosh(x) + C &&\int \frac{1}{\sqrt{a^2+x^2}} , dx = \text{arsinh}\left(\frac{x}{a}\right) + C \\ &\int \cosh(x) , dx = \sinh(x) + C &&\int \frac{1}{\sqrt{x^2-a^2}} , dx = \text{arcosh}\left(\frac{x}{a}\right) + C\\ &\int \tanh(x) \ dx = \ln(\cosh(x)) + C &&\int \frac{1}{1-x^2} , dx = \text{artanh}(x) + C \end{aligned}$$

Flashcards

#Maths/Topics/Trigonometry/Hyperbolic, #z_Legacy/Maths/A-Levels/Further-Core/Hyperbolic-Trig

What are the exponential definitions for the hyperbolic trig functions? ? $$ \sinh(x) \equiv \frac{e^x - e^{-x}}{2} \quad \quad \cosh(x) \equiv \frac{e^x + e^{-x}}{2}\quad \quad \tanh(x) \equiv \frac{\sinh(x)}{\cosh(x)} = \frac{e^{2x} - 1}{e^{2x} + 1} $$ $\begin{aligned}\text{cosech}(x) \equiv &\frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}} \quad \quad\text{sech}(x) \equiv \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \&\coth(x) \equiv \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} =\frac{e^{2x} + 1}{e^{2x} - 1}\end{aligned}$

What is Osborn's Rule for hyperbolic trig identities? ? Generally given a trig identity you can find the corresponding hyperbolic identity with Osborn's Rule $\cos(x) \rightarrow \cosh(x) \quad \sin(x) \rightarrow \sinh(x) \quad \sin^2(x) \rightarrow -\sinh^2(x) \quad\sin A \sin B \rightarrow -\sinh(A)\sinh(B)$

What are the range, domain and parities of the hyperbolic trig functions? ? Parity

• $\sinh(x)$ is odd, so $\sinh(-x) = -\sinh(x), a \in \mathbb{R}$
• $\cosh(x)$ is even, so $\cosh(-x) = \cosh(x), a \in \mathbb{R}$
• $\tanh(x)$ is odd, so $\tanh(-x) = -\tanh(x), a \in \mathbb{R}$ Range and Domain
• $\sinh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}$
• $\cosh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}, y \geq 1$
• $\tanh(x)$ - $x \in \mathbb{R}, y \in \mathbb{R}, -1 < y < 1$