Question: User: Location A has the coordinates of $42^\circ$ south $120^\circ$ east. Location B and C lie on the same parallel of latitude as location A. B is east of location A and C is west of location A. The angular distance from A to B is the same as the angular distance from A to C. The shortest distance between any 2 locations is 2000 km which is the distance from B to C. Rounding to the nearest degree, determine the coordinates of locations B and C.
1. **Problem statement:**
We have location A at $42^\circ$ south latitude and $120^\circ$ east longitude. Locations B and C lie on the same latitude as A ($42^\circ$ south), with B east of A and C west of A. The angular distance from A to B equals the angular distance from A to C. The shortest distance between B and C is 2000 km. We need to find the coordinates of B and C.
2. **Key concepts and formula:**
- Locations B and C share the same latitude as A: $\phi = -42^\circ$ (south is negative latitude).
- Longitude of A is $\lambda_A = 120^\circ$ east.
- Let the angular distance from A to B (and A to C) be $\theta$ degrees.
- Then longitude of B is $\lambda_B = 120^\circ + \theta$ and longitude of C is $\lambda_C = 120^\circ - \theta$.
3. **Distance along a parallel:**
The shortest distance $d$ between two points on the same latitude separated by longitude difference $\Delta \lambda$ is:
$$
d = R \cos(\phi) \times \Delta \lambda \times \frac{\pi}{180}
$$
where $R = 6371$ km is Earth's radius, $\phi$ is latitude in degrees, and $\Delta \lambda$ is longitude difference in degrees.
4. **Apply to B and C:**
- The longitude difference between B and C is $\Delta \lambda = (120 + \theta) - (120 - \theta) = 2\theta$ degrees.
- Given $d = 2000$ km, we have:
$$
2000 = 6371 \times \cos(42^\circ) \times 2\theta \times \frac{\pi}{180}
$$
5. **Solve for $\theta$:**
Calculate $\cos(42^\circ)$:
$$
\cos(42^\circ) \approx 0.7431
$$
Rewrite equation:
$$
2000 = 6371 \times 0.7431 \times 2\theta \times \frac{\pi}{180}
$$
Calculate constant factor:
$$
6371 \times 0.7431 \times \frac{\pi}{180} \approx 6371 \times 0.7431 \times 0.0174533 \approx 82.66
$$
So:
$$
2000 = 82.66 \times 2\theta = 165.32 \theta
$$
Therefore:
$$
\theta = \frac{2000}{165.32} \approx 12.1^\circ
$$
6. **Coordinates of B and C:**
- $\lambda_B = 120^\circ + 12.1^\circ = 132.1^\circ$ east
- $\lambda_C = 120^\circ - 12.1^\circ = 107.9^\circ$ east
- Latitude for both is $42^\circ$ south
7. **Final answer (rounded to nearest degree):**
- Location B: $42^\circ$ south, $132^\circ$ east
- Location C: $42^\circ$ south, $108^\circ$ east