Consider the equation ρ=1−cosϕ
Write this equation in rectangular coordinates.

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!

Consider the equation ρ = 1 - cos(ϕ). Write this equation in rectangular coordinates.

Explore how to convert the polar equation ρ = 1 - cos(ϕ) into rectangular coordinates step-by-step. This problem involves understanding the relationship between polar and Cartesian coordinates, and is suitable for high school students (Grades 10-12).

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