To find a polar equation for the Cartesian equation given by y², we start by recalling the relationship between Cartesian and polar coordinates. The conversion formulas are:
- x = r cos(θ)
- y = r sin(θ)
Since the equation in question is y², we can substitue the polar form of y into this expression:
Replacing y with r sin(θ) gives us:
y² = (r sin(θ))² = r² sin²(θ)Thus, the equation y² becomes:
r² sin²(θ) = y²Now, if we want to express this entirely in terms of r and θ, we can say:
If the equation is simply y², it resembles an equation in Cartesian coordinates. To represent curves in polar coordinates, we would usually need another condition or relation, as y² alone defines a family of curves (specifically, lines or parabolas depending on the context).
However, to adhere to the given input strictly and formulate:
r = ±y/sin(θ)This indicates that the relationship involves both positive and negative values, extending along the line defined by the Cartesian pole.
Therefore, while y² may represent an underlying geometric shape, we conclude with:
r^2 = y^2/sin^2(θ)This relationship gives us a starting point if further context about the curve is provided. Without additional details on the specific representation of y², this formulation stands as the foundational polar equation.