Several surprising analogies arise when comparing hyperbolic sine and cosine to their trigonometric counterparts.

Moreover, being the hyperbolic sine and cosine defined through exponential functions, they are also closely related to the complex definitions of sine and cosine, which can be regarded as a generalization of the real functions.

By an appropriate choice of the angle $z \in \mathbb{C}$, two remarkable real results can be obtained: $\sin (z) \geq 1$ (with a complex $z$ and some consequences for the cosine of the same angle) and $\cos (z) \geq 1$ (with a pure imaginary $z$).

A quick overview about these peculiarities is presented in the following document, along with several plots:

Hyperbolic functions and complex angles Hyperbolic functions and complex angles (click to download the pdf document).

Update: December 6th, 2018

Table of contents:

  • Hyperbolic sine and cosine
  • Complex sine and cosine
  • Real values greater than unity
    • Method 1
    • Method 2
  • Pure imaginary values

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