Polar coordinates question!?

Question

In 3r=sin A, r and A represent polar coordinates. Write each polar equation as an equation in rectangular coordinates (x, y).

Answer 1

x = r cos(A), y = r sin(A)
We can eliminate A to get the parametric equations of the curve in terms of r only
x = r sqt(1 - sin(a)^2) = r sqrt(1 - (ln 3r)^2)
y = r ln 3r
Oh, you want an equation in x and y only, ok...
sin(A) = y/r = y/sqrt(x^2+y^2) = ln3r = ln 3 + (1/2)ln(x^2 + y^2)
so
ln 3 + (1/2)ln(x^2 + y^2) = y/sqrt(x^2 + y^2)
Another form of the solution is
3sqrt(x^2 + y^2) = e^[y/sqrt(x^2 + y^2)]
or
9(x^2 + y^2) = e^[2y/sqrt(x^2 + y^2)]

Answer 2

We know that x = r cos A and y = r sin A, From these we find that x^2 + y^2 = (r^2)((cos A)^2 + (sin A)^2) = r^2. Now multiply your original equation by r to get 3(r^2) = r*sin A, so
3(x^2 + y^2) = y. That is the equation in x and y.