1. **Problem (i): Solve for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ the equation $\tan^2(2x + \frac{\pi}{4}) = 3$.** 2. The formula used is $\tan^2 \theta = k$ which implies $\tan \theta = \pm \sqrt{k}$. Here, $\theta = 2x + \frac{\pi}{4}$ and $k=3$. 3. Taking square root on both sides: $$\tan(2x + \frac{\pi}{4}) = \pm \sqrt{3}$$ 4. Recall that $\tan \alpha = \sqrt{3}$ at $\alpha = \frac{\pi}{3} + n\pi$ and $\tan \alpha = -\sqrt{3}$ at $\alpha = -\frac{\pi}{3} + n\pi$ for any integer $n$. 5. So, $$2x + \frac{\pi}{4} = \frac{\pi}{3} + n\pi \quad \text{or} \quad 2x + \frac{\pi}{4} = -\frac{\pi}{3} + n\pi$$ 6. Solve for $x$: $$2x = \frac{\pi}{3} - \frac{\pi}{4} + n\pi = \frac{4\pi - 3\pi}{12} + n\pi = \frac{\pi}{12} + n\pi$$ $$x = \frac{\pi}{24} + \frac{n\pi}{2}$$ and $$2x = -\frac{\pi}{3} - \frac{\pi}{4} + n\pi = -\frac{4\pi + 3\pi}{12} + n\pi = -\frac{7\pi}{12} + n\pi$$ $$x = -\frac{7\pi}{24} + \frac{n\pi}{2}$$ 7. Now find all $x$ in $(-\frac{\pi}{2}, \frac{\pi}{2})$ by testing integer values of $n$: For $x = \frac{\pi}{24} + \frac{n\pi}{2}$: - $n=0$: $x=\frac{\pi}{24} \approx 0.131$ (valid) - $n=-1$: $x=\frac{\pi}{24} - \frac{\pi}{2} = \frac{\pi}{24} - \frac{12\pi}{24} = -\frac{11\pi}{24} \approx -1.445$ (less than $-\frac{\pi}{2} \approx -1.571$, valid) - $n=1$: $x=\frac{\pi}{24} + \frac{\pi}{2} = \frac{13\pi}{24} \approx 1.701$ (greater than $\frac{\pi}{2}$, invalid) For $x = -\frac{7\pi}{24} + \frac{n\pi}{2}$: - $n=0$: $x = -\frac{7\pi}{24} \approx -0.916$ (valid) - $n=1$: $x = -\frac{7\pi}{24} + \frac{\pi}{2} = -\frac{7\pi}{24} + \frac{12\pi}{24} = \frac{5\pi}{24} \approx 0.654$ (valid) - $n=-1$: $x = -\frac{7\pi}{24} - \frac{\pi}{2} = -\frac{7\pi}{24} - \frac{12\pi}{24} = -\frac{19\pi}{24} \approx -2.487$ (less than $-\frac{\pi}{2}$, invalid) 8. Final solutions for (i) in radians: $$x \approx -1.445, -0.916, 0.131, 0.654$$ --- 9. **Problem (ii): Solve for $0 < \theta < 360^\circ$ the equation $(2 \sin \theta - \cos \theta)^2 = 1$.** 10. Taking square root: $$2 \sin \theta - \cos \theta = \pm 1$$ 11. Solve each case separately: Case 1: $2 \sin \theta - \cos \theta = 1$ Case 2: $2 \sin \theta - \cos \theta = -1$ 12. Rearrange each to express in terms of $\tan \theta$: Divide both sides by $\cos \theta$ (where $\cos \theta \neq 0$): Case 1: $$\frac{2 \sin \theta}{\cos \theta} - 1 = \frac{1}{\cos \theta}$$ $$2 \tan \theta - 1 = \sec \theta$$ But this is complicated; better to use substitution or rewrite as: Rewrite original equation: $$(2 \sin \theta - \cos \theta)^2 = 1$$ $$4 \sin^2 \theta - 4 \sin \theta \cos \theta + \cos^2 \theta = 1$$ Use $\sin^2 \theta + \cos^2 \theta = 1$: $$4 \sin^2 \theta - 4 \sin \theta \cos \theta + \cos^2 \theta = 1$$ $$4 \sin^2 \theta - 4 \sin \theta \cos \theta + (1 - \sin^2 \theta) = 1$$ $$3 \sin^2 \theta - 4 \sin \theta \cos \theta = 0$$ 13. Factor out $\sin \theta$: $$\sin \theta (3 \sin \theta - 4 \cos \theta) = 0$$ 14. So either: $$\sin \theta = 0$$ or $$3 \sin \theta - 4 \cos \theta = 0$$ 15. Solve $\sin \theta = 0$ for $0 < \theta < 360^\circ$: $$\theta = 180^\circ$$ 16. Solve $3 \sin \theta = 4 \cos \theta$: $$\tan \theta = \frac{4}{3}$$ 17. Find $\theta$: $$\theta = \tan^{-1} \left( \frac{4}{3} \right) \approx 53.1^\circ$$ Since tangent is positive in Q1 and Q3: $$\theta_1 = 53.1^\circ, \quad \theta_2 = 53.1^\circ + 180^\circ = 233.1^\circ$$ 18. Now solve case 2: $2 \sin \theta - \cos \theta = -1$ Rewrite: $$2 \sin \theta = \cos \theta - 1$$ Square both sides to avoid sign issues: $$(2 \sin \theta - \cos \theta)^2 = 1$$ But this is the original equation, so solutions come from the same factorization. Alternatively, solve directly: $$2 \sin \theta - \cos \theta = -1$$ Rearranged: $$2 \sin \theta = \cos \theta - 1$$ Divide both sides by $\cos \theta$ (where $\cos \theta \neq 0$): $$2 \tan \theta = 1 - \sec \theta$$ This is complicated; better to use substitution or test values. Alternatively, use the factorization from step 13, which covers all solutions. 19. Final solutions for (ii) to one decimal place: $$\theta = 53.1^\circ, 180.0^\circ, 233.1^\circ$$ --- **Final answers:** (i) $x \approx -1.445, -0.916, 0.131, 0.654$ radians (ii) $\theta \approx 53.1^\circ, 180.0^\circ, 233.1^\circ$