If t is any real number, then the point (cos(t),sin(t)) lies on the circleCan someone please explain to me why the parametric coordinates can only represent one branch of the rectangular hyperbola with the equation , and what would the parametric coordinates of the other branch be?

because .

For this reason sine and cosine are circular functions. So, in the same light,

for any real number t the point (cosh(t),sinh(t)) lies on the curve

because .

The other side is just (-cosh(t), -sinh(t))

This curve is called a hyperbola and therefore sinh and cosh are called hyperbolic functions.