The conversion equation is r(cos theta, sin theta)=(x, y)r(cosθ,sinθ)=(x,y), giving
r = sqrt(x^2+y^2)r=√x2+y2 (principal square root) >=0≥0,
cos theta = x/sqrt(x^2+y^2) and sin theta =y/sqrt(x^2+y^2)cosθ=x√x2+y2andsinθ=y√x2+y2.
Here, r>=r≥minimum (8csc theta)=8(8cscθ)=8, for theta in (0, pi)θ∈(0,π) .
Now, r=8csc theta =8/sin theta=8r/yr=8cscθ=8sinθ=8ry.
Cancelling non-zero r,
y=8y=8
Interestingly,
as theta to 0_+, csc theta to ooθ→0+,cscθ→∞ and,
as theta toθ→ 0_, csc to -oo_,csc→−∞.
Likewise, there is irremovable infinite discontinuity at theta=piθ=π.
So, it is proper to state that
y = 8, x in (-oo, oo)y=8,x∈(−∞,∞), in cartesian form, is equivalent to
r=8 csc theta, theta in (0. pi)r=8cscθ,θ∈(0.π), in polar form.
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