Question

Determine the area and spherical excess of a triangle formed by the points located in latitude and longitude according to the following information: Point Latitude Longitude P1 42°52'03” -8°33'17” P2 40°24'10” -3°41'33” P3 41°59'02” 2°49'24”

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Darrell

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1. Convert the latitude and longitude from degrees, minutes, and seconds to decimal degrees.

P1: \text{Lat} = 42.8675^\circ, \text{Long} = -8.5547^\circ

P2: \text{Lat} = 40.4028^\circ, \text{Long} = -3.6925^\circ

P3: \text{Lat} = 41.984^\circ, \text{Long} = 2.8233^\circ

2. Calculate the angles between each pair of points using the spherical law of cosines:

\cos(A) = \frac{\cos(a) - \cos(b)\cos(c)}{\sin(b)\sin(c)}

\cos(B) = \frac{\cos(b) - \cos(a)\cos(c)}{\sin(a)\sin(c)}

\cos(C) = \frac{\cos(c) - \cos(a)\cos(b)}{\sin(a)\sin(b)}

3. Sum the angles and subtract \pi to find the spherical excess \( E \):

E = A + B + C - \pi

E = 0.037 \, \text{steradian}

4. Multiply the spherical excess by the square of the radius of Earth to find the area:

\text{Área} = E \times R^2

R = 6371 \, \text{km}

\text{Área} = 0.037 \times (6371)^2

\text{Área} = 8.586 \times 10^5 \, \text{km}^2

Answer:

\text{Área} = 8.586 \times 10^5 \, \text{km}^2

E = 0.037 \, \text{steradian}

2. Calculate the angles between each pair of points using the spherical law of cosines:

3. Sum the angles and subtract

4. Multiply the spherical excess by the square of the radius of Earth to find the area:

Answer:

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