Convert the polar equation r=(8)/(3sintheta)+7cos(theta) to a rectangular equation

To convert the given polar equation:

\[
r = \frac{8}{3\sin\theta + 7\cos\theta}
\]

into a rectangular equation, we will use the following standard polar-to-rectangular conversion formulas:

- \( r = \sqrt{x^2 + y^2} \)
- \( x = r\cos\theta \)
- \( y = r\sin\theta \)
- \( \cos\theta = \frac{x}{r} \)
- \( \sin\theta = \frac{y}{r} \)

Now, let's follow these steps:

### Step 1: Multiply both sides of the equation by the denominator
\[
r(3\sin\theta + 7\cos\theta) = 8
\]

### Step 2: Replace \(\sin\theta\) and \(\cos\theta\) with their rectangular forms
We use the identities:
\[
\sin\theta = \frac{y}{r} \quad \text{and} \quad \cos\theta = \frac{x}{r}
\]
Substitute these into the equation:
\[
r \left( 3\frac{y}{r} + 7\frac{x}{r} \right) = 8
\]

### Step 3: Simplify the equation
Cancel out the \(r\) terms on both sides of the equation:
\[
3y + 7x = 8
\]

### Final Result:
The rectangular equation corresponding to the given polar equation is:
\[
3y + 7x = 8
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How do we convert rectangular equations back to polar form?
2. What happens when the polar equation involves higher powers of \(r\)?
3. How do we plot polar equations on a graph?
4. Can polar-to-rectangular conversion be used for 3D coordinates?
5. What applications use polar coordinates over rectangular ones?

**Tip:** When converting between coordinate systems, remember the fundamental trigonometric identities and how they relate to \(x\), \(y\), and \(r\).

Learn how to convert the polar equation r = 8 / (3 sin(theta) + 7 cos(theta)) into its rectangular form with a detailed, step-by-step explanation. Understand key formulas, such as r = sqrt(x^2 + y^2) and x = r * cos(theta), and apply polar-to-rectangular conversions. Suitable for students in Grades 10-12.

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