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. $$x^2+(i·y)^2=1\medspace →\medspace x^2-y^2=1$$

5.7.2 Hyperbolic Common Identities 🔧

Right Angle Identities

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

Sum & Difference Identities

$$\sinh(x\pm y)=\sinh(x)\cdot\cosh(y)\pm\cosh(x)\cdot\sinh(y)$$ $$\cosh(x\pm y)=\cosh(x)\cdot\cosh(y)\pm\sinh(x)\cdot\sinh(y)$$

5.7.3 Limits of Hyperbolic Functions 🔧

5.7.4 Derivatives of Hyperbolic Functions 🔧

5.7.5 Real Version of Euler's Formula 🔧

5.7.6 Analytic Hyperbolic Functions 🔧

5.7.7 Analytic Hyperbolic Inverses 🔧