5.7 Contents

1. Unit Hyperbola Definitions 🔧

2. Hyperbolic Common Identities 🔧

3. Limits of Hyperbolic Functions 🔧

4. Derivatives of Hyperbolic Functions 🔧

5. Real Version of Euler's Formula 🔧

6. Analytic Hyperbolic Functions 🔧

7. Analytic Hyperbolic Inverses 🔧

5.7.1 Unit Hyperbola Definitions 🔧

Hyperbolic trig functions operate similarly as the trig functions on the unit circle, except they neither rotate nor have periodicity. $$\sinh(\theta)=y/h\qquad\text{csch}(\theta)=h/y$$ $$\cosh(\theta)=x/h\qquad\text{sech}(\theta)=h/x$$ $$\tanh(\theta)=y/x\qquad\coth(\theta)=x/y$$ $$\theta=\sinh^{-1}(y/h)\qquad\theta=\text{csch}^{-1}(h/y)$$ $$\theta=\cosh^{-1}(x/h)\qquad\theta=\text{sech}^{-1}(h/x)$$ $$\theta=\tanh^{-1}(y/x)\qquad\theta=\coth^{-1}(x/y)$$

Relations

A unit circle rotated one half radian on the unit double cone along its tangential axis results in the unit hyperbola. The number $1$ rotated one half radian in the complex plane results the imaginary unit $i$. The insertion of $i$ into the unit circle equation results in the unit hyperbola equation. $$x^2+(i\cdot y)^2=1\medspace\to\medspace x^2-y^2=1$$

5.7.2 Hyperbolic Common Identities 🔧

Right Angle Identities

$$\sinh^2(\theta)=\cosh^2(\theta)+1$$ $$\tanh^2(\theta)+\text{sech}^2(\theta)=1$$ $$\coth^2(\theta)=\text{csch}^2(\theta)+1$$

Sum & Difference Identities

$$\sinh(a\pm b)=\sinh(a)\cdot\cosh(b)\pm\cosh(a)\cdot\sinh(b)$$ $$\cosh(a\pm b)=\cosh(a)\cdot\cosh(b)\pm\sinh(a)\cdot\sinh(b)$$

5.7.3 Limits of Hyperbolic Functions 🔧

$$\lim\limits_{\theta\to 0}\frac{\sinh(\theta)}{\theta}=1\qquad\lim\limits_{\theta\to 0}\frac{\cosh(\theta)-1}{\theta}=0$$

5.7.4 Derivatives of Hyperbolic Functions 🔧

$$\frac{d}{dx}\sinh(\theta)=\cosh(\theta)$$ $$\frac{d}{dx}\cosh(\theta)=\sinh(\theta)$$ $$\frac{d}{dx}\text{sech}(\theta)=-\text{sech}(\theta)\cdot\tanh(\theta)$$ $$\frac{d}{dx}\text{csch}(\theta)=-\text{csch}(\theta)\cdot\coth(\theta)$$ $$\frac{d}{dx}\tanh(\theta)=\text{sech}^2(\theta)$$ $$\frac{d}{dx}\coth(\theta)=-\text{csch}^2(\theta)$$

5.7.5 Real Version of Euler's Formula 🔧

$$e^\theta=\cosh(\theta)+\sinh(\theta)$$

Proof

5.7.6 Analytic Hyperbolic Functions 🔧

$$\sinh(\theta)=-i\cdot\sin(i\cdot\theta)=\frac{e^\theta-e^{-\theta}}{2}$$ $$\cosh(\theta)=\cos(i\cdot\theta)=\frac{e^\theta+e^{-\theta}}{2}$$ $$\text{sech}(\theta)=\sec(i\cdot\theta)=\frac{2}{e^\theta+e^{-\theta}}$$ $$\text{csch}(\theta)=i\cdot\csc(i\cdot\theta)=\frac{2}{e^\theta-e^{-\theta}}$$ $$\tanh(\theta)=-i\cdot\tan(i\cdot\theta)=\frac{e^\theta-e^{-\theta}}{e^\theta+e^{-\theta}}$$ $$\coth(\theta)=i\cdot\cot(i\cdot\theta)=\frac{e^\theta+e^{-\theta}}{e^\theta-e^{-\theta}}$$

Proof of Sine and Cosine

Proof of Secant, Cosecant, Tangent, and Cotangent

5.7.7 Analytic Hyperbolic Inverses 🔧

$$\sinh^{-1}(\theta)=\ln\big(\theta+\sqrt{\theta^2+1}\big),\medspace\forall\theta$$ $$\cosh^{-1}(\theta)=\ln\big(\theta+\sqrt{\theta^2-1}\big),\medspace\theta\ge1$$ $$\text{sech}^{-1}(\theta)=\ln\big(\theta^{-1}+\sqrt{\theta^{-2}-1}\big),\medspace 0\lt\theta\lt1$$ $$\text{csch}^{-1}(\theta)=\ln\big(\theta^{-1}+\sqrt{\theta^{-2}+1}\big),\medspace\theta\ne0$$ $$\tanh^{-1}(\theta)=\frac{1}{2}\cdot\frac{1+\theta}{1-\theta},\medspace|\theta|\lt1$$ $$\coth^{-1}(\theta)=\frac{1}{2}\cdot\frac{\theta+1}{\theta-1},\medspace|\theta|\gt1$$

Proof of Sine

Proof of Cosine

Proof of Secant

Proof of Cosecant

Proof of Tangent

Proof of Cotangent