Converting equations from polar to rectangular form involves a systematic approach, as there isn't a single operation that applies universally. In polar coordinates, points are represented by \( r \) (the radius) and \( \theta \) (the angle), while in rectangular coordinates, they are represented by \( x \) and \( y \). The key relationships to remember are:

1. \( x = r \cos(\theta) \)

2. \( y = r \sin(\theta) \)

3. \( r^2 = x^2 + y^2 \)

To convert a polar equation to rectangular form, the first step is to manipulate the equation to include one of these terms. For example, consider the polar equation \( r = 4 \). Since it does not contain any of the necessary terms, we can square both sides to obtain:

\( r^2 = 16 \)

Next, we replace \( r^2 \) with \( x^2 + y^2 \), leading to:

\( x^2 + y^2 = 16 \)

This equation represents a circle with a radius of 4, centered at the origin, and is already in standard form.

In another example, for the polar equation \( r = \sec(\theta) \), we can rewrite the secant function as:

\( r = \frac{1}{\cos(\theta)} \)

To eliminate the fraction, multiply both sides by \( \cos(\theta) \):

\( r \cos(\theta) = 1 \)

Substituting \( r \cos(\theta) \) with \( x \) gives:

\( x = 1 \

This represents a vertical line at \( x = 1 \), which is also in standard form.

For the polar equation \( r = 6 \sin(\theta) \), we start by multiplying both sides by \( r \) to obtain:

\( r^2 = 6r \sin(\theta) \

Substituting gives:

\( x^2 + y^2 = 6y \

To rewrite this in standard form, we rearrange it to:

\( x^2 + y^2 - 6y = 0 \

Next, we complete the square for the \( y \) terms. The expression \( y^2 - 6y \) can be transformed into a perfect square by adding 9:

\( x^2 + (y - 3)^2 = 9 \

This represents a circle centered at \( (0, 3) \) with a radius of 3, now in standard form.

By following these steps and strategies, you can effectively convert polar equations into rectangular form and identify their geometric representations. Practice with various equations will enhance your understanding and proficiency in this conversion process.