1. The problem is to analyze the equation $$2\sqrt{18-18\cos\theta} = 6\pi - 3\theta$$ and find a way to approach it. 2. First, recognize that the left side involves a square root of an expression with cosine, and the right side is a linear expression in $\theta$. 3. The formula inside the square root can be simplified by factoring: $$\sqrt{18 - 18\cos\theta} = \sqrt{18(1 - \cos\theta)}$$ 4. Recall the trigonometric identity: $$1 - \cos\theta = 2\sin^2\left(\frac{\theta}{2}\right)$$ 5. Substitute this identity into the expression: $$\sqrt{18 \cdot 2 \sin^2\left(\frac{\theta}{2}\right)} = \sqrt{36 \sin^2\left(\frac{\theta}{2}\right)}$$ 6. Simplify the square root: $$\sqrt{36} \cdot \sqrt{\sin^2\left(\frac{\theta}{2}\right)} = 6 \left|\sin\left(\frac{\theta}{2}\right)\right|$$ 7. So the equation becomes: $$2 \times 6 \left|\sin\left(\frac{\theta}{2}\right)\right| = 6\pi - 3\theta$$ 8. Simplify the left side: $$12 \left|\sin\left(\frac{\theta}{2}\right)\right| = 6\pi - 3\theta$$ 9. Divide both sides by 3: $$\cancel{3} \times 4 \left|\sin\left(\frac{\theta}{2}\right)\right| = \cancel{3} \times (2\pi - \theta)$$ 10. This simplifies to: $$4 \left|\sin\left(\frac{\theta}{2}\right)\right| = 2\pi - \theta$$ 11. The hint is to now solve for $\theta$ by considering the behavior of the sine function and the linear term on the right side. 12. You can analyze the equation graphically or numerically to find solutions for $\theta$. This approach simplifies the original equation and sets you up to solve it effectively.