Question

In problem graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=csct,y=cott;\frac{\mathrm{\pi }}{4}\le t\le \frac{\mathrm{\pi }}{2}.$

Short answer

The equation is$x^2-y^2=1.$

Step-by-step solution

Step 1. Given information.

$x=csct,y=cott$ is given.

Step 2. Form the graph and equation.

we form the rectangular equation by eliminating t in the equation-

$$\begin{gathered} x=csct \\ y=cott \\ cs{c}^{2}t=1+co{t}^{2}t \\ {x}^2=1+y^2 \\ {x}^2-y^2=1. \end{gathered}$$

Graph the curve defined by the parametric equations -

$t$	$x$	$y$	$(x,y)$
$\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$	$\sqrt{2}$	$1$	$\left(\sqrt{2},1\right)$
$\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$1$	$0$	$\left(1,0\right)$