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The problem in the image asks to "Find the initial course, final course, and great circle distance from 75° 45'N, 030° 46'E to 40° 00'N, 110° 15'E."

To solve this problem, we need to use the formulas related to great-circle navigation, which typically involve spherical trigonometry. Here's a step-by-step approach for calculating each part:

### 1. **Initial Course (Bearing)**

The initial course (also known as the bearing) is the angle between the north direction and the direction of travel. To calculate the initial course between two points on a sphere, you can use:

\[
\text{Initial Course} = \text{atan2}(\sin(\Delta \lambda) \cdot \cos(\phi_2), \cos(\phi_1) \cdot \sin(\phi_2) - \sin(\phi_1) \cdot \cos(\phi_2) \cdot \cos(\Delta \lambda))
\]

Where:
- \(\phi_1, \phi_2\) are the latitudes of the first and second points, respectively.
- \(\lambda_1, \lambda_2\) are the longitudes of the first and second points.
- \(\Delta \lambda = \lambda_2 - \lambda_1\)

### 2. **Final Course**

The final course can be found by swapping the positions of the two points and recalculating the bearing from the destination to the origin. Another way to calculate it is using the reverse of the initial course formula after the great-circle distance is known.

### 3. **Great Circle Distance**

The great circle distance is the shortest path between two points on the surface of a sphere. The formula for this is:

\[
\Delta \sigma = \cos^{-1}(\sin(\phi_1) \cdot \sin(\phi_2) + \cos(\phi_1) \cdot \cos(\phi_2) \cdot \cos(\Delta \lambda))
\]

Where \(\Delta \sigma\) is the angular distance in radians. To convert this into a linear distance, multiply by the radius \(R\) of the Earth (mean radius = 6,371 km):

\[
\text{Distance} = R \cdot \Delta \sigma
\]

#### Given Data:
- Latitude 1: \(75^\circ 45'N = 75.75^\circ\)
- Longitude 1: \(30^\circ 46'E = 30.767^\circ\)
- Latitude 2: \(40^\circ 00'N = 40^\circ\)
- Longitude 2: \(110^\circ 15'E = 110.25^\circ\)

Using these formulas, we can calculate the initial course, final course, and the great circle distance.

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Find the initial course, final course, and great circle distance from 75° 45'N, 030° 46'E to 40° 00'N, 110° 15'E.

Learn how to calculate the initial course, final course, and great circle distance between 75° 45'N, 030° 46'E and 40° 00'N, 110° 15'E using spherical trigonometry. This detailed solution includes key formulas and navigation concepts.

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