Convert the the polar equation $x = cos θ$ $x=cos theta$ and $y = sec θ$ $y = sec theta$ where $θ \in [0, \frac{π}{2}]$ $theta in [0,pi/2]$ to cartesian coordinates?

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Answer 1 · 2017-06-30T21:19:58

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

We have:

$x = cos θ$ $x=cos theta$ where $θ \in [0, \frac{π}{2}]$ $theta in [0,pi/2]$
$y = sec θ$ $y = sec theta$

First let us graph the curve for validation purposes:
enter image source here

So to convert to rectangular coordinates we can use:

$y = sec θ \Rightarrow y = \frac{1}{cos θ}$ $y = sec theta => y = 1/cos theta$
$:. y = 1/x$

If we sketch this graph we have:
enter image source here

So we have the correct graph but an incorrect domain.

$theta = 0 \ \ => x=cos0 = 1$
$theta = pi/2 => x=cos(pi/2) = 0$

And if we ow restrict the domain of $y = 1/x$ to $x in [0,1]$ we have:
enter image source here

Convert the the polar equation x=cos theta x=cosθ x=cos theta and y = sec theta y=secθ y = sec theta where theta in [0,pi/2]θ∈[0,π2]theta in [0,pi/2] to cartesian coordinates?