1. **Problem statement:** Solve the equation $2 \cos^3 x - 1 = 0$ for $x \in [0, 2\pi]$, expressing solutions in terms of $\pi$ radians. 2. **Formula and rules:** Recall that $\cos^3 x = (\cos x)^3$. We want to isolate $\cos x$ and then find all $x$ values in the given interval that satisfy the equation. 3. **Step-by-step solution:** 1. Start with the equation: $$2 \cos^3 x - 1 = 0$$ 2. Add 1 to both sides: $$2 \cos^3 x = 1$$ 3. Divide both sides by 2: $$\cancel{2} \cos^3 x = \frac{1}{\cancel{2}}$$ $$\cos^3 x = \frac{1}{2}$$ 4. Take the cube root of both sides: $$\cos x = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}$$ 5. Numerically, $\frac{1}{\sqrt[3]{2}} \approx 0.7937$. 6. Find all $x$ in $[0, 2\pi]$ such that $\cos x = 0.7937$. Since cosine is positive in Quadrants I and IV, the solutions are: $$x = \pm \arccos(0.7937) + 2k\pi$$ For $k=0$ and $x \in [0, 2\pi]$: - First solution: $x_1 = \arccos(0.7937) \approx 0.646 \text{ radians}$ - Second solution: $x_2 = 2\pi - 0.646 = 5.637 \text{ radians}$ 7. Express in terms of $\pi$: $$x_1 \approx 0.646 \approx \frac{0.646}{\pi} \pi \approx 0.206 \pi$$ $$x_2 \approx 5.637 \approx \frac{5.637}{\pi} \pi \approx 1.795 \pi$$ 4. **Final answer:** $$x = 0.206\pi, \quad 1.795\pi$$ These are the two solutions to the equation $2 \cos^3 x - 1 = 0$ on the interval $[0, 2\pi]$.