How would you transform the polar equation "r csc theta = 2" to an equation in rectangular coordinates? Answer needs to have an x and a y on one side. Equations to use: x = r cos theta; y = r sin theta; x^2 + y^2 = r^2

Question

How would you transform the polar equation "r csc theta = 2" to an equation in rectangular coordinates?  Answer needs to have an x and a y on one side.  Equations to use: x = r cos theta; y = r sin theta; x^2 + y^2 = r^2

dumbcow · Answer

csc = 1/sin

rewrite equation
$$r = 2 \sin \theta$$
square both sides to get an r^2 term
$$r^2 = 4 \sin^2 \theta$$
substitute --->  r^2 = x^2+y^2   and   sin = y/r
$$x^2 + y^2 = 4(\frac{y^2}{r^2})$$
substitute for r^2 again
$$x^2 +y^2 = \frac{4y^2}{x^2 + y^2}$$

dumbcow · Answer

wait sorry we can simply it further
$$(x^2 +y^2)^2 = 4y^2$$
sqrt both sides
$$x^2 +y^2 = 2y$$

now you have a circle

anonymous · Answer

Thank you so much for the response, unfortunately that is not the answer my teacher was looking for.  The correct answer was $$x^2+(y-1)^2=1$$  just in case somebody out there gets a similar question.

dumbcow · Answer

same thing just in different forms

$$x^2 +y^2 = 2y$$
$$x^2 + (y^2 -2y) = 0$$
$$x^2 + (y^2 -2y +1) = 1$$
$$x^2 + (y-1)^2 = 1$$