1. **Problem (i): Solve $2 \sin \theta = 1$ for $\theta$.\n\n2. Divide both sides by 2: $$\cancel{2} \sin \theta = \frac{1}{\cancel{2}} \implies \sin \theta = \frac{1}{2}.$$\n\n3. Use the inverse sine function: $$\theta = \sin^{-1}\left(\frac{1}{2}\right).$$\n\n4. From the unit circle, $\sin \theta = \frac{1}{2}$ at $\theta = 30^\circ$ (principal value).\n\n5. **Answer (i):** $\theta = 30^\circ$.\n\n---\n\n1. **Problem (ii): Solve $5 \cos \theta = 2$ for $\theta$.\n\n2. Divide both sides by 5: $$\cancel{5} \cos \theta = \frac{2}{\cancel{5}} \implies \cos \theta = \frac{2}{5} = 0.4.$$\n\n3. Use the inverse cosine function: $$\theta = \cos^{-1}(0.4).$$\n\n4. Calculate $\theta$: $\theta \approx 66^\circ$ (nearest degree).\n\n5. **Answer (ii):** $\theta = 66^\circ$.\n\n---\n\n1. **Problem (iii): Solve $2 \tan \theta = 1$ for $\theta$.\n\n2. Divide both sides by 2: $$\cancel{2} \tan \theta = \frac{1}{\cancel{2}} \implies \tan \theta = \frac{1}{2} = 0.5.$$\n\n3. Use the inverse tangent function: $$\theta = \tan^{-1}(0.5).$$\n\n4. Calculate $\theta$: $\theta \approx 26^\circ$ (nearest degree).\n\n5. **Answer (iii):** $\theta = 26^\circ$.\n\n---\n\n1. **Problem 13: Find angle $x$ in a right triangle with vertical side 12 m and horizontal side 30 m.\n\n2. The angle $x$ is between the shadow (adjacent side) and the hypotenuse. We use the tangent function: $$\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{30} = 0.4.$$\n\n3. Use inverse tangent: $$x = \tan^{-1}(0.4).$$\n\n4. Calculate $x$: $x \approx 22^\circ$ (nearest degree).\n\n5. **Answer:** $x = 22^\circ$.