1. **Problem (i): Solve for $0 < \theta \leq 360^\circ$ the equation $4 \tan \theta + 5 \sin \theta = 0$.** 2. **Recall the definitions:** - $\tan \theta = \frac{\sin \theta}{\cos \theta}$. 3. **Rewrite the equation:** $$4 \tan \theta + 5 \sin \theta = 0 \implies 4 \frac{\sin \theta}{\cos \theta} + 5 \sin \theta = 0$$ 4. **Factor out $\sin \theta$:** $$\sin \theta \left( \frac{4}{\cos \theta} + 5 \right) = 0$$ 5. **Set each factor to zero:** - $\sin \theta = 0$ - $\frac{4}{\cos \theta} + 5 = 0$ 6. **Solve $\sin \theta = 0$ for $0 < \theta \leq 360^\circ$:** - $\theta = 0^\circ, 180^\circ, 360^\circ$ - Exclude $0^\circ$ since $0 < \theta$, so solutions: $180^\circ, 360^\circ$ 7. **Solve $\frac{4}{\cos \theta} + 5 = 0$:** $$\frac{4}{\cos \theta} = -5 \implies \cos \theta = -\frac{4}{5} = -0.8$$ 8. **Find $\theta$ where $\cos \theta = -0.8$ in $0 < \theta \leq 360^\circ$:** - $\theta = \cos^{-1}(-0.8)$ - Using a calculator: $\theta \approx 143.1^\circ$ (2nd quadrant) - Also, $\theta = 360^\circ - 143.1^\circ = 216.9^\circ$ (3rd quadrant) 9. **Final solutions for (i):** $$\boxed{180^\circ, 216.9^\circ, 360^\circ, 143.1^\circ}$$ --- 10. **Problem (ii): Solve for $0 < x < \pi$ the equation $$\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{5}{\cos x}$$** 11. **Rewrite terms:** $$\tan x + \cot x = \frac{5}{\cos x}$$ 12. **Express $\tan x + \cot x$ as a single fraction:** $$\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}$$ 13. **Substitute back:** $$\frac{1}{\sin x \cos x} = \frac{5}{\cos x}$$ 14. **Multiply both sides by $\sin x \cos x$:** $$1 = 5 \sin x$$ 15. **Solve for $\sin x$:** $$\sin x = \frac{1}{5} = 0.2$$ 16. **Find $x$ in $0 < x < \pi$ where $\sin x = 0.2$:** - $x = \sin^{-1}(0.2) \approx 0.201$ radians - Second solution in $(0, \pi)$ is $x = \pi - 0.201 = 2.940$ radians 17. **Final solutions for (ii) to 3 significant figures:** $$\boxed{0.201, 2.94}$$