Hyperbolic Functions

Hyperbolic Sine

$$\sinh x=\frac{1}{2}(e^x-e^{-x})$$

• Domain: $x\in (-\infty ,\infty )$
• Range: $y\in (-\infty ,\infty )$
• Odd function

Hyperbolic Cos

$$\cosh x=\frac{1}{2}(e^x+e^{-x})$$

• Domain: $x\in (-\infty ,\infty )$
• Range: $y\ge 1$
• Even function

Hyperbolic Tan

$$\begin{array}{rl}\tanh x& =\frac{e^x-e^{-x}}{e^x+e^{-x}}\\ & =\frac{e^{2x}-1}{e^{2x}+1}\\ & =\frac{1-e^{-2x}}{1+e^{-2x}}\end{array}$$