1. **State the problem:** Solve the trigonometric equation $$3 + 5 \sin \theta = 1$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Isolate the sine term:** $$3 + 5 \sin \theta = 1$$ Subtract 3 from both sides: $$5 \sin \theta = 1 - 3$$ $$5 \sin \theta = -2$$ Divide both sides by 5: $$\sin \theta = \frac{-2}{5}$$ Show cancellation: $$\sin \theta = \cancel{5} \frac{-2}{\cancel{5}} = -\frac{2}{5}$$ 3. **Find the reference angle:** Calculate the inverse sine of the positive value: $$\theta_r = \sin^{-1}\left(\frac{2}{5}\right) \approx 23.58^\circ$$ 4. **Determine the quadrants:** Since $$\sin \theta$$ is negative, $$\theta$$ lies in Quadrants III and IV. 5. **Calculate the solutions:** - Quadrant III: $$\theta = 180^\circ + \theta_r = 180^\circ + 23.58^\circ = 203.58^\circ$$ - Quadrant IV: $$\theta = 360^\circ - \theta_r = 360^\circ - 23.58^\circ = 336.42^\circ$$ 6. **Final answer:** $$\theta = 203.6^\circ \text{ and } 336.4^\circ$$ --- **Additional problem:** Simplify the ratio 384 km : 192 km. 1. Divide both terms by their greatest common divisor (GCD), which is 192: $$\frac{384}{192} : \frac{192}{192} = 2 : 1$$ **Final simplified ratio:** $$2 : 1$$