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Let's work on converting the given polar equations into rectangular (Cartesian) coordinates.

### Key Polar-Rectangular Coordinate Relations:
1. \( x = r \cos(\theta) \)
2. \( y = r \sin(\theta) \)
3. \( r^2 = x^2 + y^2 \)
4. \( \tan(\theta) = \frac{y}{x} \)
5. \( \sin(\theta) = \frac{y}{r} \)
6. \( \cos(\theta) = \frac{x}{r} \)

We will now apply these relationships to each equation.

### (a) \( r = 6 \sin(2\theta) \)

Using the trigonometric identity for double angle, \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we rewrite the equation:
\[
r = 6 \cdot 2 \sin(\theta) \cos(\theta)
\]
\[
r = 12 \sin(\theta) \cos(\theta)
\]

Now, express \( \sin(\theta) \) and \( \cos(\theta) \) in terms of \( x \), \( y \), and \( r \):
\[
\sin(\theta) = \frac{y}{r}, \quad \cos(\theta) = \frac{x}{r}
\]

Substitute these into the equation:
\[
r = 12 \cdot \frac{y}{r} \cdot \frac{x}{r}
\]
Multiply both sides by \( r^2 \) to eliminate the denominator:
\[
r^3 = 12xy
\]
Now, use \( r^2 = x^2 + y^2 \):
\[
(x^2 + y^2)^{3/2} = 12xy
\]
This is the rectangular form of the given polar equation.

### (b) \( r = \frac{2}{1 + 3 \cos(\theta)} \)

Substitute \( \cos(\theta) = \frac{x}{r} \):
\[
r = \frac{2}{1 + 3 \cdot \frac{x}{r}}
\]

Multiply both sides by \( 1 + \frac{3x}{r} \) to eliminate the denominator:
\[
r \left( 1 + \frac{3x}{r} \right) = 2
\]
Simplify:
\[
r + 3x = 2
\]
Substitute \( r = \sqrt{x^2 + y^2} \):
\[
\sqrt{x^2 + y^2} + 3x = 2
\]
This is the rectangular form of the second polar equation.

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Let me know if you need further clarification or details on these transformations.

### 5 Related Questions:
1. How do polar and rectangular coordinates relate geometrically?
2. What are the advantages of using polar coordinates over rectangular coordinates in certain problems?
3. Can we graph the rectangular forms of these equations?
4. How would we convert a general polar equation \( r = f(\theta) \) into rectangular coordinates?
5. What trigonometric identities are most commonly used in polar-rectangular conversions?

### Tip:
Always double-check whether multiplying by \( r \) or \( r^2 \) simplifies the equation before using more complicated transformations.

Encuentre una ecuaciones rectangulares, para las ecuaciones polares dadas: (a) r = 6sen(2θ), (b) r = 2 / (1 + 3cos(θ))

Learn how to convert polar equations to rectangular form. This tutorial covers key polar and trigonometric relationships to convert equations like r = 6sen(2θ) and r = 2 / (1 + 3cos(θ)) into Cartesian coordinates. Suitable for high school and college students.

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