1. **Problem Statement:** We have a right triangle formed by a planet, its moon, and its sun. The distance between the planet and its moon is 141,000 km, and the angle at the moon is 80.4°. We need to find the distance between the moon and the sun. 2. **Identify the triangle sides and angles:** - The right angle is at the planet. - The side between the planet and moon (vertical leg) is 141,000 km. - The angle at the moon is 80.4°. - We want the distance between the moon and the sun, which is the hypotenuse opposite the right angle. 3. **Use trigonometric relationships:** Since the right angle is at the planet, the side between planet and moon is adjacent to the angle at the moon. We use the cosine function: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ Here, $\theta = 80.4^\circ$, adjacent side = 141,000 km, hypotenuse = distance from moon to sun (let's call it $d$). 4. **Set up the equation:** $$\cos(80.4^\circ) = \frac{141000}{d}$$ 5. **Solve for $d$:** $$d = \frac{141000}{\cos(80.4^\circ)}$$ 6. **Calculate $\cos(80.4^\circ)$:** Using a calculator, $$\cos(80.4^\circ) \approx 0.165$$ 7. **Calculate $d$:** $$d = \frac{141000}{0.165} \approx 854545.45$$ 8. **Round to the nearest whole number:** $$d \approx 854545$$ km **Final answer:** The distance between the moon and the sun is approximately **854,545 km**.