In 3r = sin A, r and A represent polar coordinates...?

Question

How do I write each polar equation as an equation in rectangular coordinates (x,y) ?

Answer 1

Given polar coordinates (r,A), find Cartesian coordinates (x,y).

Remember the identities.
x = rcosA
y = rsinA
x² + y² = r²
___________

3r = sinA
3r = y/r
3r² = y

3(x² + y²) = y
3x² + 3y² - y = 0
x² + y² - y/3 = 0

Complete the square.

x² + (y² - y/3 + 1/36) = 1/36
x² + (y - 1/6)² = 1/36

This is a circle with center (0, 1/6) and radius 1/6.

Answer 2

If (x,y) are the cartesian co-ordinates corresponding to the polar co-ordinates (r,A), then:
r^2 = x^2 + y^2 .....(1)
x = r cos(A) ........(2)
y = r sin(A) ........(3)

3r = sin(A)
Substituting for sin(A) from (3):
3r = y / r
3r^2 = y
Substituting for r^2 from (1):
3(x^2 + y^2) = y.