To convert the equation $x + 3y + 6 = 0$ into polar form, we use the relationships between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$, which are:

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

Step-by-Step Conversion:

1. Start with the original equation: $x + 3y + 6 = 0$

2. Substitute $x = r \cos \theta$ and $y = r \sin \theta$: $r \cos \theta + 3r \sin \theta + 6 = 0$

3. Factor out $r$: $r(\cos \theta + 3 \sin \theta) = -6$

4. Solve for $r$: $r = \frac{-6}{\cos \theta + 3 \sin \theta}$

Thus, the polar form of the equation is:

$r = \frac{-6}{\cos \theta + 3 \sin \theta}$

Would you like further explanation on how the polar form was derived, or do you have any other questions?

Related Questions:

1. How do you convert any general Cartesian equation to polar form?
2. What is the significance of the polar coordinate system in mathematics?
3. How does the polar coordinate system simplify equations of circles or lines?
4. Can you show how polar and Cartesian coordinates relate geometrically?
5. What are common applications of polar coordinates in physics?

Tip: In polar coordinates, the angle $\theta$ represents the direction of the point from the origin, while $r$ represents the distance from the origin. This system is useful when analyzing problems with rotational symmetry.