To convert polar equations to rectangular form, we use the following identities that relate polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$:

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

Now, let's solve each part.

(a) $r \sin \theta = -3$

Using the identity $y = r \sin \theta$, we can rewrite the equation as: $y = -3$ This is the rectangular form.

(b) $r = 4$

Using the identity $r^2 = x^2 + y^2$, square both sides of the equation: $r^2 = 4^2 = 16$ So, the rectangular form is: $x^2 + y^2 = 16$ This represents a circle with a radius of 4 centered at the origin.

Would you like more details or clarifications on these steps?

Here are five related questions to expand on this topic:

1. How do we convert rectangular coordinates to polar coordinates?
2. What is the general form of a polar equation for a line?
3. How can you graph the equation $x^2 + y^2 = 16$?
4. What are some common polar-to-rectangular conversions for different curves?
5. How does the unit circle help in converting between polar and rectangular forms?

Tip: Whenever converting between coordinate systems, it's crucial to remember the core identities that link polar coordinates ($r, \theta$) to rectangular coordinates ($x, y$). These identities simplify the transformation process.