Hyperbolic functions

Need to know

What you need to know

• $\cosh^2(x) - \sinh^2(x) = 1$
• $1 - \tanh^2(x) = \text{sech}^2(x)$ (Divide the first identity by $\cosh^2(x)$)
• $\coth^2(x) - 1 = \text{cosech}^2(x)$ (Divide the first identity by $\sinh^2(x)$)

Explanation

The Exponential Cousins of Trigonometry

1. Learn the definitions of sinh(x) and cosh(x) using e^x and e^-x.
2. Derive the definitions for tanh(x) and the reciprocal functions (sech, cosech, coth).
3. Master the core identity cosh²(x) - sinh²(x) = 1 and its variations.
4. Sketch the graphs, paying close attention to key points, symmetry, and asymptotes.