Hyperbolic Functions Cheatsheet

Cheatsheet Content

1. Introduction to Hyperbolic Functions Hyperbolic functions are analogues of the ordinary trigonometric functions (also called circular functions) but defined using the hyperbola $x^2 - y^2 = 1$ instead of the circle $x^2 + y^2 = 1$. They are fundamental in various areas of mathematics, physics, and engineering, especially in describing catenary curves, special relativity, and linear differential equations. 2. Definitions and Comparison with Circular Functions Circular functions are defined using a unit circle. Hyperbolic functions are defined using the unit hyperbola and exponential functions. 2.1. Circular Functions (Unit Circle $x^2 + y^2 = 1$) $\cos \theta = x$ $\sin \theta = y$ $\tan \theta = \frac{y}{x}$ Parameterization: $x = \cos \theta$, $y = \sin \theta$ 2.2. Hyperbolic Functions (Unit Hyperbola $x^2 - y^2 = 1$) Defined in terms of the natural exponential function $e^x$: Hyperbolic Cosine (cosh): $\cosh x = \frac{e^x + e^{-x}}{2}$ Hyperbolic Sine (sinh): $\sinh x = \frac{e^x - e^{-x}}{2}$ Hyperbolic Tangent (tanh): $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ Hyperbolic Cotangent (coth): $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ (for $x \ne 0$) Hyperbolic Secant (sech): $\text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$ Hyperbolic Cosecant (csch): $\text{csch } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$ (for $x \ne 0$) Parameterization: $x = \cosh t$, $y = \sinh t$ traces the right branch of the hyperbola $x^2 - y^2 = 1$. 3. Key Identities Many hyperbolic identities are analogous to circular identities, often differing by a sign. 3.1. Fundamental Identities Circular: $\cos^2 x + \sin^2 x = 1$ Hyperbolic: $\cosh^2 x - \sinh^2 x = 1$ (This is the most fundamental difference!) Circular: $1 + \tan^2 x = \sec^2 x$ Hyperbolic: $1 - \tanh^2 x = \text{sech}^2 x$ Circular: $\cot^2 x + 1 = \csc^2 x$ Hyperbolic: $\coth^2 x - 1 = \text{csch}^2 x$ 3.2. Addition Formulas Circular: $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$ $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$ $\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}$ Hyperbolic: $\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$ $\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$ $\tanh(x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}$ 3.3. Double Angle Formulas Circular: $\sin(2x) = 2 \sin x \cos x$ $\cos(2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$ Hyperbolic: $\sinh(2x) = 2 \sinh x \cosh x$ $\cosh(2x) = \cosh^2 x + \sinh^2 x = 2 \cosh^2 x - 1 = 1 + 2 \sinh^2 x$ 4. Derivatives Derivatives of hyperbolic functions are very similar to circular functions, often without the sign changes. $\frac{d}{dx}(\sinh x) = \cosh x$ $\frac{d}{dx}(\cosh x) = \sinh x$ $\frac{d}{dx}(\tanh x) = \text{sech}^2 x$ $\frac{d}{dx}(\coth x) = -\text{csch}^2 x$ $\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x$ $\frac{d}{dx}(\text{csch } x) = -\text{csch } x \coth x$ Compare with circular derivatives: $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ 5. Integrals $\int \sinh x \, dx = \cosh x + C$ $\int \cosh x \, dx = \sinh x + C$ $\int \text{sech}^2 x \, dx = \tanh x + C$ $\int \text{csch}^2 x \, dx = -\coth x + C$ $\int \text{sech } x \tanh x \, dx = -\text{sech } x + C$ $\int \text{csch } x \coth x \, dx = -\text{csch } x + C$ 6. Inverse Hyperbolic Functions These are the inverse functions of the hyperbolic functions, expressed in terms of logarithms. $\text{arsinh } x = \ln(x + \sqrt{x^2+1})$ $\text{arccosh } x = \ln(x + \sqrt{x^2-1})$, for $x \ge 1$ $\text{arctanh } x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$, for $|x| $\text{arcoth } x = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$, for $|x| > 1$ $\text{arsech } x = \ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right)$, for $0 $\text{arcsch } x = \ln\left(\frac{1}{x} + \frac{\sqrt{1+x^2}}{|x|}\right)$, for $x \ne 0$ 7. Relationship with Complex Exponentials (Euler's Formula) Euler's formula connects circular functions with complex exponentials: $e^{i\theta} = \cos \theta + i \sin \theta$. $\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ $\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$ Notice the striking similarity to the definitions of $\cosh x$ and $\sinh x$. This suggests a deep connection: $\cosh x = \cos(ix)$ $\sinh x = -i \sin(ix)$ $\cos x = \cosh(ix)$ $\sin x = -i \sinh(ix)$ These relations show that hyperbolic functions are essentially circular functions of imaginary arguments. 8. Graphs and Properties 8.1. $\cosh x$ Domain: $(-\infty, \infty)$ Range: $[1, \infty)$ Even function: $\cosh(-x) = \cosh x$ Graph resembles a parabola or a catenary curve (shape of a hanging chain). Minimum at $(0,1)$. 8.2. $\sinh x$ Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$ Odd function: $\sinh(-x) = -\sinh x$ Passes through the origin $(0,0)$. 8.3. $\tanh x$ Domain: $(-\infty, \infty)$ Range: $(-1, 1)$ Odd function: $\tanh(-x) = -\tanh x$ Asymptotes at $y=1$ and $y=-1$. Passes through the origin $(0,0)$.