Convert the polar equation r^2 sin(2theta)=6 to rectangular form

To convert the polar equation \( r^2 \sin(2\theta) = 6 \) to rectangular form, we'll use the following polar-to-rectangular relationships:

1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)
4. \( \sin(2\theta) = 2 \sin \theta \cos \theta = \frac{2xy}{r^2} \)

Now, let's work through the conversion:

### Step 1: Rewrite \( r^2 \sin(2\theta) = 6 \) using the double-angle identity
\[
r^2 \sin(2\theta) = r^2 \cdot \frac{2xy}{r^2}
\]
This simplifies to:
\[
2xy = 6
\]

### Step 2: Final form in rectangular coordinates
Now we can divide both sides by 2:
\[
xy = 3
\]

So, the rectangular form of the given polar equation is:
\[
xy = 3
\]

Would you like more details, or have any other questions?

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Here are 5 related questions to deepen your understanding:
1. How would the conversion change if the equation were \( r^2 \cos(2\theta) = 6 \)?
2. What is the geometric interpretation of the equation \( xy = 3 \)?
3. How would you convert a rectangular equation back to polar form?
4. Can we use other polar relationships to derive the same result?
5. What types of curves are typically expressed in polar coordinates?

**Tip:** Double-angle identities, like \( \sin(2\theta) = 2 \sin \theta \cos \theta \), are often essential in converting polar equations to rectangular form.

Convert the polar equation r^2 sin(2theta) = 6 to rectangular form

Discover how to convert the polar equation r^2 sin(2theta) = 6 to rectangular form using trigonometric identities and coordinate transformations. This solution is ideal for high school students studying coordinate conversions between polar and rectangular forms.

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