To convert the given cylindrical equation $\theta = \frac{\pi}{6}$ into rectangular coordinates, we can use the relationships between cylindrical and rectangular coordinates:

1. In cylindrical coordinates: $x = r \cos \theta$ $y = r \sin \theta$

2. Since $\theta = \frac{\pi}{6}$, we know the angle is fixed.

Given $\theta = \frac{\pi}{6}$, we can use the fact that: $\tan \theta = \frac{y}{x}$ Thus: $\tan \frac{\pi}{6} = \frac{y}{x}$ Since $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$, the equation becomes: $\frac{y}{x} = \frac{1}{\sqrt{3}}$ or equivalently: $y = \frac{1}{\sqrt{3}} x$ which can be simplified further by multiplying both sides by $\sqrt{3}$: $\sqrt{3} y = x$

This line, $x = \sqrt{3} y$, represents the surface in rectangular coordinates.

Would you like further details or have any questions?

Here are five related questions:

1. How would you convert other values of $\theta$ to rectangular coordinates?
2. How can we represent cylindrical surfaces with variable $\theta$?
3. What if we had a different angle, like $\theta = \frac{\pi}{4}$?
4. How does this conversion apply in three-dimensional spaces?
5. Can we extend this to spherical coordinates?

Tip: Remember, for converting cylindrical equations, express $\theta$ in terms of $\frac{y}{x}$ for simpler transformations to rectangular coordinates.