linguistic:paraphrasing, structural:scholar_edition
cosine

Using the hyperbolic sine and cosine functions cosh, sinh {\displaystyle \cosh,\sinh }, a parametric representation of the hyperbola x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1} can be obtained, which is similar to the parametric representation of an ellipse: (± a cosh t, b sinh t), t ∈ R , {\displaystyle (\pm a\cosh t,b\sinh t),\,t\in \mathbb {R} \,} which satisfies the Cartesian equation because cosh 2 t − sinh 2 t = 1. {\displaystyle \cosh ^{2}t-\sinh ^{2}t=1.}

Further parametric representations are given in the section Parametric equations below.