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 $r$.