Math Problem Statement

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

Solution

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:

Initial Course=atan2(sin(Δλ)cos(ϕ2),cos(ϕ1)sin(ϕ2)sin(ϕ1)cos(ϕ2)cos(Δλ))\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:

  • ϕ1,ϕ2\phi_1, \phi_2 are the latitudes of the first and second points, respectively.
  • λ1,λ2\lambda_1, \lambda_2 are the longitudes of the first and second points.
  • Δλ=λ2λ1\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:

Δσ=cos1(sin(ϕ1)sin(ϕ2)+cos(ϕ1)cos(ϕ2)cos(Δλ))\Delta \sigma = \cos^{-1}(\sin(\phi_1) \cdot \sin(\phi_2) + \cos(\phi_1) \cdot \cos(\phi_2) \cdot \cos(\Delta \lambda))

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

Distance=RΔσ\text{Distance} = R \cdot \Delta \sigma

Given Data:

  • Latitude 1: 7545N=75.7575^\circ 45'N = 75.75^\circ
  • Longitude 1: 3046E=30.76730^\circ 46'E = 30.767^\circ
  • Latitude 2: 4000N=4040^\circ 00'N = 40^\circ
  • Longitude 2: 11015E=110.25110^\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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Math Problem Analysis

Mathematical Concepts

Great Circle Navigation
Spherical Trigonometry
Bearing and Distance Calculations

Formulas

Initial Course = atan2(sin(Δλ) * cos(φ2), cos(φ1) * sin(φ2) - sin(φ1) * cos(φ2) * cos(Δλ))
Great Circle Distance Δσ = cos^(-1)(sin(φ1) * sin(φ2) + cos(φ1) * cos(φ2) * cos(Δλ))
Distance = R * Δσ

Theorems

Spherical Law of Cosines

Suitable Grade Level

University level, particularly for navigation or advanced trigonometry courses.