The given equation is:

$\rho = 1 - \cos \phi$

This is expressed in polar coordinates $(\rho, \phi)$, where $\rho$ represents the radial distance and $\phi$ is the polar angle. To convert this to rectangular coordinates $(x, y)$, we use the following relationships between polar and rectangular coordinates:

$x = \rho \cos \phi, \quad y = \rho \sin \phi, \quad \rho = \sqrt{x^2 + y^2}, \quad \text{and} \quad \cos \phi = \frac{x}{\sqrt{x^2 + y^2}}$

Step-by-Step Conversion

1. Substitute $\rho$ and $\cos \phi$: $\rho = 1 - \cos \phi$ Using $\cos \phi = \frac{x}{\sqrt{x^2 + y^2}}$, we get: $\rho = 1 - \frac{x}{\sqrt{x^2 + y^2}}$

2. Substitute $\rho = \sqrt{x^2 + y^2}$: $\sqrt{x^2 + y^2} = 1 - \frac{x}{\sqrt{x^2 + y^2}}$

3. Multiply both sides by $\sqrt{x^2 + y^2}$ to eliminate the denominator: $(\sqrt{x^2 + y^2})^2 = (1 - \frac{x}{\sqrt{x^2 + y^2}})(\sqrt{x^2 + y^2})$ $x^2 + y^2 = \sqrt{x^2 + y^2} - x$

4. Square both sides to eliminate the square root on the right-hand side: $(x^2 + y^2)(x^2 + y^2) = (1 - x)^2$

5. Expand and simplify:

This would give the final rectangular form.

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Would you like more details on any part of the process?

Here are some follow-up questions to consider:

1. What are polar coordinates and how do they differ from rectangular coordinates?
2. Can every polar equation be transformed into rectangular coordinates?
3. How would you graph the equation $\rho = 1 - \cos \phi$ in polar coordinates?
4. What types of curves do such polar equations typically represent?
5. How do sinusoidal functions influence the shape of polar curves?

Tip: Always keep track of the relationship between polar and rectangular coordinates when transforming equations!