1. **State the problem:** Solve the equation $$4 \tan \theta + 5 \sin \theta = 0$$ for $$0 < \theta \leq 360^\circ$$. 2. **Recall the definitions and formulas:** - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$. - The equation can be rewritten using this identity. 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 equal to zero:** - $$\sin \theta = 0$$ - $$\frac{4}{\cos \theta} + 5 = 0$$ 6. **Solve $$\sin \theta = 0$$ for $$0 < \theta \leq 360^\circ$$:** - $$\sin \theta = 0$$ at $$\theta = 0^\circ, 180^\circ, 360^\circ$$. - Since $$0 < \theta \leq 360^\circ$$, valid solution is $$\theta = 180^\circ$$. 7. **Solve $$\frac{4}{\cos \theta} + 5 = 0$$:** - Multiply both sides by $$\cos \theta$$ (note $$\cos \theta \neq 0$$): $$4 + 5 \cos \theta = 0$$ - Rearrange: $$5 \cos \theta = -4$$ - Divide: $$\cos \theta = -\frac{4}{5} = -0.8$$ 8. **Find $$\theta$$ where $$\cos \theta = -0.8$$ in $$0 < \theta \leq 360^\circ$$:** - $$\cos \theta = -0.8$$ occurs in the second and third quadrants. - Use inverse cosine: $$\theta = \cos^{-1}(-0.8)$$ - Calculate: $$\theta_1 = 143.1^\circ$$ (second quadrant) $$\theta_2 = 216.9^\circ$$ (third quadrant) 9. **List all solutions:** $$\theta = 180^\circ, 143.1^\circ, 216.9^\circ$$ **Final answer:** $$\boxed{\theta = 143.1^\circ, 180^\circ, 216.9^\circ}$$