1. **State the problem:** Solve the equation $$4(\sin x)^2 - 2 = 0$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:** $$4(\sin x)^2 - 2 = 0$$ Add 2 to both sides: $$4(\sin x)^2 = 2$$ Divide both sides by 4: $$\cancel{4}(\sin x)^2 = \frac{2}{\cancel{4}}$$ Simplifies to: $$ (\sin x)^2 = \frac{1}{2} $$ 3. **Take the square root of both sides:** $$ \sin x = \pm \sqrt{\frac{1}{2}} = \pm \frac{\sqrt{2}}{2} $$ 4. **Find all solutions for $$x$$ in $$[0, 2\pi)$$ where $$\sin x = \pm \frac{\sqrt{2}}{2}$$:** - For $$\sin x = \frac{\sqrt{2}}{2}$$, solutions are: $$ x = \frac{\pi}{4}, \frac{3\pi}{4} $$ - For $$\sin x = -\frac{\sqrt{2}}{2}$$, solutions are: $$ x = \frac{5\pi}{4}, \frac{7\pi}{4} $$ 5. **Write the solutions in increasing order:** $$ x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $$ 6. **Answer the question about the missing numbers:** The missing numbers in the green boxes are 5 and 7. **Final answer:** $$ x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $$