To rewrite the given Cartesian equation $y = 6x^2$ as a polar equation, we can use the polar coordinate relationships:

$x = r\cos\theta \quad \text{and} \quad y = r\sin\theta$

Steps:

1. Substitute $x = r\cos\theta$ and $y = r\sin\theta$ into $y = 6x^2$: $r\sin\theta = 6(r\cos\theta)^2$

2. Expand $(r\cos\theta)^2$: $r\sin\theta = 6r^2\cos^2\theta$

3. Simplify to isolate $r$: Divide through by $r$ (assuming $r \neq 0$): $\sin\theta = 6r\cos^2\theta$

4. Solve for $r$: $r = \frac{\sin\theta}{6\cos^2\theta}$

Thus, the polar equation is: $r(\theta) = \frac{\sin\theta}{6\cos^2\theta}$

Let me know if you would like further clarification or more examples!

Related Questions:

1. How do you convert polar equations back to Cartesian equations?
2. What is the relationship between $\tan\theta$ and Cartesian coordinates $x, y$?
3. Can you rewrite $x^2 + y^2 = 1$ in polar form?
4. What happens when $r = 0$ in a polar equation?
5. How do $\sin\theta$ and $\cos\theta$ affect the shape of polar graphs?

Tip:

When converting equations, always substitute $x = r\cos\theta$ and $y = r\sin\theta$, and don't forget the identity $x^2 + y^2 = r^2$ for simplifications!