Question

Find a parametrization of the curve \(x^2-y^2=1\) indicating the range of allowed values of your parametrization.
Please, help

Answer 1

@amoodarya

Answer 2

@Michele_Laino

Answer 3

I think that a possible parametrization can be this:
\[\Large \left\{ \begin{gathered}
x = \cosh \theta \hfill \\
y = \sinh \theta \hfill \\
\end{gathered} \right.,\quad \theta \in {\mathbf{R}}\]

Answer 4

how about x= sec t
y = tan t?

Answer 5

please keep in mind that the subsequent condition, holds:
\[\Large {\left( {\cosh \theta } \right)^2} - {\left( {\sinh \theta } \right)^2} = 1\]

Answer 6

I know hyper trigs are work. However, if I don't know, how to derive to it?

Answer 7

I prefer hyperbolic function, since, your function is an hyperbola, and it is their natural application

Answer 8

There must be something to restrict tan and sec, but I don't know it. Since it is nothing wrong with tan^2 -sec^2 =1

Answer 9

I think that your parametrization also works, nevertheless it is the first time that I see it

Answer 10

One more question: I know the form is a hyperbola. What is the link between hyperbolic function with a hyperbola?

Answer 11

note that :
\[|x| \geq 1\]

Answer 12

@amoodarya how do you know?

Answer 13

Oh, I got it @amoodarya

Answer 14

the requested link is:
suppose to have your hyperbola:
x^2-y^2=1
then we want to compute the area of the hyperbolic sector: