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:
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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#