1. **State the problem:** Solve the equation $$2 \sin t \cos t + \sin t - 2 \cos t - 1 = 0$$ for $$t$$ in the interval $$[0, 2\pi]$$, where the solutions must be multiples of $$\pi$$ expressed as fractions or integers. 2. **Rewrite the equation:** Group terms to factor: $$2 \sin t \cos t + \sin t - 2 \cos t - 1 = (2 \sin t \cos t + \sin t) - (2 \cos t + 1) = 0$$ 3. **Factor each group:** $$\sin t (2 \cos t + 1) - (2 \cos t + 1) = 0$$ 4. **Factor out the common term:** $$(2 \cos t + 1)(\sin t - 1) = 0$$ 5. **Set each factor equal to zero:** - $$2 \cos t + 1 = 0$$ - $$\sin t - 1 = 0$$ 6. **Solve each equation:** - For $$2 \cos t + 1 = 0$$: $$2 \cos t = -1$$ $$\cos t = -\frac{1}{2}$$ $$t = \frac{2\pi}{3}, \frac{4\pi}{3}$$ (in $$[0, 2\pi]$$) - For $$\sin t - 1 = 0$$: $$\sin t = 1$$ $$t = \frac{\pi}{2}$$ (in $$[0, 2\pi]$$) 7. **Final solutions:** $$t = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}$$ 8. **Express answers as multiples of $$\pi$$:** $$t = \frac{1}{2}\pi, \frac{2}{3}\pi, \frac{4}{3}\pi$$ **Answer:** $$\frac{1}{2}, \frac{2}{3}, \frac{4}{3}$$