How do you convert $r = 2csctheta$ into cartesian form?

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Answer 1 · 2015-11-24T08:45:12

$y = 2$

When converting between polar and rectangular coordinates, we can use the following equalities:
${(r^2 = x^2 + y^2), (tan(theta) = y/x):}$

(To see why these equalities hold, there is a good explanation provided here: How do you convert rectangular coordinates to polar coordinates? )

From the first equality
$r = sqrt(x^2 + y^2)$

From the second equality, we can consider the following right triangle:
enter image source here

To obtain $csc(theta) = sqrt(x^2 + y^2)/y$

Putting that together, we get

$sqrt(x^2 + y^2) = 2sqrt(x^2+y^2)/y$

$=> 1 = 2/y$

$=> y = 2$

How do you convert r = 2cscthetar = 2csctheta into cartesian form?