To convert the polar equation $r = \cos \theta + \sin \theta$ into rectangular form, we need to use the relationships between polar and rectangular coordinates.

Recall the following equations:

1. $x = r \cos \theta$
2. $y = r \sin \theta$
3. $r^2 = x^2 + y^2$

Step-by-Step Conversion:

1. Start with the given equation in polar form: $r = \cos \theta + \sin \theta$

2. Multiply both sides by $r$ to eliminate $\theta$: $r^2 = r \cos \theta + r \sin \theta$

3. Now substitute $r \cos \theta = x$ and $r \sin \theta = y$ into the equation: $r^2 = x + y$

4. Using $r^2 = x^2 + y^2$, substitute this on the left side: $x^2 + y^2 = x + y$

5. Rearranging this equation gives the rectangular form: $x^2 + y^2 - x - y = 0$

Thus, the rectangular form of the equation $r = \cos \theta + \sin \theta$ is: $x^2 + y^2 - x - y = 0$

Would you like more details or have any questions?

Related questions:

1. How do you graph the equation $x^2 + y^2 - x - y = 0$?
2. What is the significance of polar to rectangular conversions in geometry?
3. Can this conversion method be used for any polar equation?
4. How does this conversion change for other trigonometric forms in polar coordinates?
5. What is the relationship between the center of the circle and the equation in rectangular form?

Tip: When converting from polar to rectangular form, always recall the basic relations between polar coordinates and rectangular coordinates: $r \cos \theta = x$ and $r \sin \theta = y$.