How can I find a polar equation for the curve represented by the Cartesian equation y^2?

To convert a Cartesian equation into its polar form, we need to use the relationships between polar coordinates (r, θ) and Cartesian coordinates (x, y). In polar coordinates, the following equations hold:

• x = r * cos(θ)
• y = r * sin(θ)

Given the Cartesian equation y², we’ll start off by rewriting y in terms of polar coordinates:

y = r * sin(θ)

Now, substituting this expression into the given equation, we have:

y² = (r * sin(θ))²

This results in:

y² = r² * sin²(θ)

As the original equation was simply y² (which can be treated as y² = 0), we can express this in polar coordinates accordingly:

r² * sin²(θ) = 0

This equation tells us that:

• r = 0 or sin(θ) = 0

Thus, we can derive the polar equations:

• If r = 0, we have the pole (origin) as a point.
• If sin(θ) = 0, this gives us angles where θ = nπ (n is an integer), indicating all points along the x-axis.

In conclusion, the polar representation for the curve defined purely by y² is:

r = 0  or  θ = nπ

These two expressions indicate that the curve aligns with the x-axis and includes the origin as well.