1. **Problem:** Find the great circle (G/C) initial course and distance from latitude 49°56' N, longitude 006°27' W to latitude 25°58' N, longitude 077°00' W. 2. **Formula and rules:** - Convert all latitudes and longitudes to decimal degrees. - Use the spherical law of cosines for distance: $$\cos d = \sin \phi_1 \sin \phi_2 + \cos \phi_1 \cos \phi_2 \cos \Delta \lambda$$ where $d$ is the central angle in radians, $\phi_1$, $\phi_2$ are latitudes in radians, and $\Delta \lambda$ is difference in longitude in radians. - Distance $D = R \times d$, where $R$ is Earth's radius (approx. 3440 nautical miles). - Initial course $\theta$ is given by: $$\tan \theta = \frac{\sin \Delta \lambda \cos \phi_2}{\cos \phi_1 \sin \phi_2 - \sin \phi_1 \cos \phi_2 \cos \Delta \lambda}$$ 3. **Convert coordinates:** - $\phi_1 = 49 + \frac{56}{60} = 49.9333^\circ$ N - $\lambda_1 = - (6 + \frac{27}{60}) = -6.45^\circ$ (west is negative) - $\phi_2 = 25 + \frac{58}{60} = 25.9667^\circ$ N - $\lambda_2 = - (77 + 0) = -77^\circ$ Convert degrees to radians: - $\phi_1 = 49.9333 \times \frac{\pi}{180} = 0.8713$ rad - $\lambda_1 = -6.45 \times \frac{\pi}{180} = -0.1126$ rad - $\phi_2 = 25.9667 \times \frac{\pi}{180} = 0.4533$ rad - $\lambda_2 = -77 \times \frac{\pi}{180} = -1.3440$ rad 4. **Calculate $\Delta \lambda$:** $$\Delta \lambda = \lambda_2 - \lambda_1 = -1.3440 - (-0.1126) = -1.2314$$ 5. **Calculate central angle $d$:** $$\cos d = \sin 0.8713 \times \sin 0.4533 + \cos 0.8713 \times \cos 0.4533 \times \cos (-1.2314)$$ Calculate each term: - $\sin 0.8713 = 0.7661$ - $\sin 0.4533 = 0.4380$ - $\cos 0.8713 = 0.6427$ - $\cos 0.4533 = 0.8989$ - $\cos (-1.2314) = 0.3323$ So: $$\cos d = 0.7661 \times 0.4380 + 0.6427 \times 0.8989 \times 0.3323 = 0.3355 + 0.1919 = 0.5274$$ 6. **Find $d$:** $$d = \arccos(0.5274) = 1.0175 \text{ radians}$$ 7. **Calculate distance $D$:** $$D = 3440 \times 1.0175 = 3499 \text{ nautical miles (approx.)}$$ 8. **Calculate initial course $\theta$:** $$\tan \theta = \frac{\sin (-1.2314) \times \cos 0.4533}{\cos 0.8713 \times \sin 0.4533 - \sin 0.8713 \times \cos 0.4533 \times \cos (-1.2314)}$$ Calculate numerator: - $\sin (-1.2314) = -0.9435$ - $\cos 0.4533 = 0.8989$ - Numerator = $-0.9435 \times 0.8989 = -0.8483$ Calculate denominator: - $\cos 0.8713 = 0.6427$ - $\sin 0.4533 = 0.4380$ - $\sin 0.8713 = 0.7661$ - $\cos 0.4533 = 0.8989$ - $\cos (-1.2314) = 0.3323$ - Denominator = $0.6427 \times 0.4380 - 0.7661 \times 0.8989 \times 0.3323 = 0.2815 - 0.2287 = 0.0528$ 9. **Calculate $\theta$:** $$\theta = \arctan \left( \frac{-0.8483}{0.0528} \right) = \arctan(-16.07) = -86.4^\circ$$ Since denominator is positive and numerator negative, $\theta$ is in the 4th quadrant, so add 360°: $$\theta = 360 - 86.4 = 273.6^\circ$$ 10. **Interpretation:** Initial course is approximately $273.6^\circ$ (almost due west). **Final answers:** - Initial course: $273.6^\circ$ - Distance: 3499 nautical miles