# Convert the the polar equation  x=cos theta  and  y = sec theta  where theta in [0,pi/2] to cartesian coordinates?

Jun 30, 2017

$y = \frac{1}{x}$ where $x \in \left[0 , 1\right]$

#### Explanation:

We have:

$x = \cos \theta$ where $\theta \in \left[0 , \frac{\pi}{2}\right]$
$y = \sec \theta$

First let us graph the curve for validation purposes:

So to convert to rectangular coordinates we can use:

$y = \sec \theta \implies y = \frac{1}{\cos} \theta$
$\therefore y = \frac{1}{x}$

If we sketch this graph we have:

So we have the correct graph but an incorrect domain.

$\theta = 0 \setminus \setminus \implies x = \cos 0 = 1$
$\theta = \frac{\pi}{2} \implies x = \cos \left(\frac{\pi}{2}\right) = 0$

And if we ow restrict the domain of $y = \frac{1}{x}$ to $x \in \left[0 , 1\right]$ we have: