1. **State the problem:** Convert the polar equation $$r = 4\sin(2\theta)$$ to rectangular coordinates. 2. **Recall the formulas:** - Polar to rectangular conversions use $$x = r\cos\theta$$ and $$y = r\sin\theta$$. - Also, $$r^2 = x^2 + y^2$$ and $$\tan\theta = \frac{y}{x}$$. - The double-angle identity for sine is $$\sin(2\theta) = 2\sin\theta\cos\theta$$. 3. **Rewrite the given equation using the double-angle identity:** $$r = 4\sin(2\theta) = 4 \times 2\sin\theta\cos\theta = 8\sin\theta\cos\theta$$ 4. **Multiply both sides by $$r$$ to use rectangular forms:** $$r \times r = r \times 8\sin\theta\cos\theta$$ $$r^2 = 8r\sin\theta\cos\theta$$ 5. **Substitute rectangular equivalents:** $$r^2 = x^2 + y^2$$ $$r\sin\theta = y$$ $$r\cos\theta = x$$ So, $$x^2 + y^2 = 8 \times y \times x$$ 6. **Rewrite the equation:** $$x^2 + y^2 = 8xy$$ 7. **Final rectangular form:** $$x^2 + y^2 - 8xy = 0$$ This is the rectangular form of the given polar equation representing a four-petal rose centered at the origin.