1. **State the problem:** Solve the trigonometric equation $$2\cos(x) - 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi)$$. 2. **Use the formula and rules:** To solve for $$x$$, isolate $$\cos(x)$$: $$2\cos(x) - 1 = 0$$ Add 1 to both sides: $$2\cos(x) = 1$$ Divide both sides by 2: $$\cancel{2}\cos(x) = \frac{1}{\cancel{2}}$$ which simplifies to: $$\cos(x) = \frac{1}{2}$$ 3. **Find all solutions for $$\cos(x) = \frac{1}{2}$$ on $$[0, 2\pi)$$:** Recall that $$\cos(x) = \frac{1}{2}$$ at angles where the cosine value is positive one-half. This occurs at: $$x = \frac{\pi}{3}$$ and $$x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$$ 4. **Final answer:** $$x = \frac{\pi}{3}, \frac{5\pi}{3}$$ These are the two solutions in the interval $$[0, 2\pi)$$.