1. State the problem: Express $3\cos \theta - 4\sin \theta$ in the form $R\cos(\theta - \alpha)$ and solve the equation $3\cos \theta - 4\sin \theta = 0$ for $0 \leq \theta \leq 2\pi$. 2. Expressing in the form $R\cos(\theta - \alpha)$ requires: $$R\cos(\theta - \alpha) = R(\cos \theta \cos \alpha + \sin \theta \sin \alpha) = R\cos \alpha \cos \theta + R\sin \alpha \sin \theta.$$ 3. Match coefficients with $3\cos \theta - 4\sin \theta$: $$R\cos \alpha = 3,\quad R\sin \alpha = -4.$$ 4. Find $R$: $$R = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$$ 5. Find $\alpha$: $$\tan \alpha = \frac{R \sin \alpha}{R \cos \alpha} = \frac{-4}{3} = -\frac{4}{3}.$$ Since $R\cos \alpha = 3 > 0$ and $R\sin \alpha = -4 < 0$, $\alpha$ lies in the fourth quadrant. Calculate: $$\alpha = \arctan\left(-\frac{4}{3}\right) = -0.9273\text{ radians (approx)}.$$ Convert to positive angle between $0$ and $2\pi$: $$\alpha = 2\pi - 0.9273 = 5.3559\text{ radians (approx)}.$$ 6. Rewrite the expression: $$3\cos \theta - 4\sin \theta = 5\cos(\theta - 5.3559).$$ 7. Solve the equation: $$5\cos(\theta - 5.3559) = 0 \implies \cos(\theta - 5.3559) = 0.$$ 8. The solutions for $\cos x = 0$ in $[0, 2\pi]$ are: $$x = \frac{\pi}{2}, \frac{3\pi}{2}.$$ Let $x = \theta - 5.3559$, so: $$\theta - 5.3559 = \frac{\pi}{2}, \frac{3\pi}{2}.$$ 9. Find $\theta$: $$\theta = 5.3559 + \frac{\pi}{2} = 5.3559 + 1.5708 = 6.9267,$$ exceeds $2\pi$, subtract $2\pi$: $$6.9267 - 2\pi = 6.9267 - 6.2832 = 0.6435.$$ $$\theta = 5.3559 + \frac{3\pi}{2} = 5.3559 + 4.7124 = 10.0683,$$ subtract $2\pi$ twice (since above $2\pi$): $$10.0683 - 2\pi = 3.7851,$$ which lies in $[0, 2\pi]$. 10. Final solutions in $[0, 2\pi]$: $$\theta \approx 0.6435, 3.7851.$$