A branch of a hyperbola

Problem

Verify that $x(t)=\cosh t$, $y(t)=\sinh t$ ($t\in\mathbb{R}$) parametrises a branch of the hyperbola $x^2-y^2=1$.

Solution

Use the fundamental identity of the hyperbolic functions $\cosh^2 t - \sinh^2 t = 1$: $x^2 - y^2 = \cosh^2 t - \sinh^2 t = \boxed{1}.$ So every point of the curve satisfies the equation of the hyperbola. Moreover $x=\cosh t\ge 1>0$ for every $t\in\mathbb{R}$: only the right branch is traced (the one with $x\ge 1$).

Links

Topics: Complex numbers
Concepts: Implicit form · Parametrisation
Skills: Proving
Exercise type: Proof