1. **Problem:** Solve $\sin x = -\frac{\sqrt{2}}{2}$ on $[0, 2\pi)$. 2. **Formula and rules:** The sine function equals $-\frac{\sqrt{2}}{2}$ at angles where the reference angle is $\frac{\pi}{4}$ and sine is negative (third and fourth quadrants). 3. **Intermediate work:** - Reference angle: $\frac{\pi}{4}$ - Solutions in $[0, 2\pi)$ where $\sin x = -\frac{\sqrt{2}}{2}$ are: $$x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}, \quad x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}$$ 4. **Answer:** $$x = \left\{ \frac{5\pi}{4}, \frac{7\pi}{4} \right\}$$ Note: The user wrote $\{3\pi/4, 7\pi/4\}$ but $3\pi/4$ corresponds to $\sin x = +\frac{\sqrt{2}}{2}$, not negative. --- 2. **Problem:** Solve $\cos^2 x - 1 = 0$ on $[0, 2\pi)$. 3. **Formula and rules:** - Rearrange: $\cos^2 x = 1$ - Taking square root: $\cos x = \pm 1$ 4. **Intermediate work:** - $\cos x = 1$ at $x=0, 2\pi$ - $\cos x = -1$ at $x=\pi$ 5. **Answer:** $$x = \{0, \pi, 2\pi\}$$ --- 3. **Problem:** Solve $2 \cos^2 x - \cos x - 1 = 0$ on $[0, 2\pi)$. 4. **Formula and rules:** Treat as quadratic in $\cos x$: $$2u^2 - u - 1 = 0, \quad u = \cos x$$ 5. **Intermediate work:** - Factor or use quadratic formula: $$u = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm 3}{4}$$ - Solutions: - $u = \frac{1+3}{4} = 1$ - $u = \frac{1-3}{4} = -\frac{1}{2}$ 6. **Find $x$ values:** - For $\cos x = 1$, $x = 0, 2\pi$ - For $\cos x = -\frac{1}{2}$, reference angle $\frac{\pi}{3}$, solutions in second and third quadrants: $$x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}, \quad x = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$$ 7. **Answer:** $$x = \left\{0, 2\pi, \frac{2\pi}{3}, \frac{4\pi}{3} \right\}$$ --- 4. **Problem:** Solve $4 \sin^2 x - 3 = 0$ on $[0, 2\pi)$. 5. **Formula and rules:** - Rearrange: $$4 \sin^2 x = 3 \Rightarrow \sin^2 x = \frac{3}{4}$$ - Take square root: $$\sin x = \pm \frac{\sqrt{3}}{2}$$ 6. **Find $x$ values:** - Reference angle $\frac{\pi}{3}$ - $\sin x = \frac{\sqrt{3}}{2}$ at $x = \frac{\pi}{3}, \frac{2\pi}{3}$ - $\sin x = -\frac{\sqrt{3}}{2}$ at $x = \frac{4\pi}{3}, \frac{5\pi}{3}$ 7. **Answer:** $$x = \left\{ \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \right\}$$ --- 5. **Problem:** Solve $\sin x - 2 \sin x \cos x = 0$ on $[0, 2\pi)$. 6. **Formula and rules:** Factor out $\sin x$: $$\sin x (1 - 2 \cos x) = 0$$ 7. **Intermediate work:** - Set each factor to zero: - $\sin x = 0$ - $1 - 2 \cos x = 0 \Rightarrow \cos x = \frac{1}{2}$ 8. **Find $x$ values:** - $\sin x = 0$ at $x = 0, \pi, 2\pi$ - $\cos x = \frac{1}{2}$ at $x = \frac{\pi}{3}, \frac{5\pi}{3}$ 9. **Answer:** $$x = \left\{0, \pi, 2\pi, \frac{\pi}{3}, \frac{5\pi}{3} \right\}$$ --- 6. **Problem:** Solve $\cos^3 x - \cos x = 0$ on $[0, 2\pi)$. 7. **Formula and rules:** Factor: $$\cos x (\cos^2 x - 1) = 0$$ 8. **Intermediate work:** - $\cos x = 0$ - $\cos^2 x - 1 = 0 \Rightarrow \cos^2 x = 1 \Rightarrow \cos x = \pm 1$ 9. **Find $x$ values:** - $\cos x = 0$ at $x = \frac{\pi}{2}, \frac{3\pi}{2}$ - $\cos x = 1$ at $x=0, 2\pi$ - $\cos x = -1$ at $x=\pi$ 10. **Answer:** $$x = \left\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \right\}$$ --- 7. **Problem:** Solve $\cos^2 x = 1 - \sin x$ on $[0, 2\pi)$. 8. **Formula and rules:** Use identity $\cos^2 x = 1 - \sin^2 x$: $$1 - \sin^2 x = 1 - \sin x$$ 9. **Intermediate work:** - Subtract 1 from both sides: $$-\sin^2 x = -\sin x$$ - Multiply both sides by $-1$: $$\sin^2 x = \sin x$$ - Rearrange: $$\sin^2 x - \sin x = 0$$ - Factor: $$\sin x (\sin x - 1) = 0$$ 10. **Find $x$ values:** - $\sin x = 0$ at $x=0, \pi, 2\pi$ - $\sin x = 1$ at $x=\frac{\pi}{2}$ 11. **Answer:** $$x = \left\{0, \pi, 2\pi, \frac{\pi}{2} \right\}$$ --- 8. **Problem:** Solve $\sin^2 x \cos x = \cos x$ on $[0, 2\pi)$. 9. **Formula and rules:** Rearrange: $$\sin^2 x \cos x - \cos x = 0$$ 10. **Intermediate work:** - Factor out $\cos x$: $$\cos x (\sin^2 x - 1) = 0$$ - Set each factor to zero: - $\cos x = 0$ - $\sin^2 x - 1 = 0 \Rightarrow \sin^2 x = 1 \Rightarrow \sin x = \pm 1$ 11. **Find $x$ values:** - $\cos x = 0$ at $x=\frac{\pi}{2}, \frac{3\pi}{2}$ - $\sin x = 1$ at $x=\frac{\pi}{2}$ - $\sin x = -1$ at $x=\frac{3\pi}{2}$ 12. **Answer:** $$x = \left\{ \frac{\pi}{2}, \frac{3\pi}{2} \right\}$$