Solve for following applying spherical trigonometry...

1.Papua New Guinea (37° 45'N, 122°27'W)(since course was 260°) distance travelled =2250nm.
A) use law of cosine to calculate a and thus altitude:or
B) use law of sine to calculate N and thus longitude: or
C) use law of sine to calculate B and thus the course on arrival..

To solve this problem using spherical trigonometry, we can use the following steps:

A) Using the Law of Cosines, we can calculate the altitude (a) at Papua New Guinea:

1. Convert the latitude and longitude to radians:
Latitude: 37° 45' N = 37.75° = 37.75 * (π/180) = 0.65979 radians
Longitude: 122° 27' W = 122.45° = 122.45 * (π/180) = -2.13772 radians

2. Convert the course (260°) to radians:
Course: 260° = 260 * (π/180) = 4.53786 radians

3. Using the Law of Cosines formula:
cos a = cos θ1 * cos θ2 + sin θ1 * sin θ2 * cos C
where θ1 and θ2 are the latitudes, and C is the difference in longitudes.

Since we are calculating the altitude, θ1 will be the initial latitude (Papua New Guinea) and θ2 will be the final latitude.

Hence, cos a = cos 0.65979 * cos 0 + sin 0.65979 * sin 0 * cos (-2.13772)

4. Calculate the value of a by taking the inverse cosine (arccos) of that value:
a = arccos(cos a)

This will give you the altitude (a) at Papua New Guinea.

B) Using the Law of Sines, we can calculate the longitude (N) on arrival:

1. Using the Law of Sines formula:
sin N = sin C * sin θ2 / sin a
where C is the difference in longitudes, θ2 is the final latitude, and a is the altitude.

Since we are calculating the longitude, C will be the course on arrival, θ2 will be the final latitude, and a will be the altitude.

Hence, sin N = sin 4.53786 * sin θ2 / sin a

2. Calculate the value of N by taking the inverse sine (arcsin) of that value:
N = arcsin(sin N)

This will give you the longitude (N) on arrival.

C) Using the Law of Sines, we can calculate the course on arrival (B):

1. Using the Law of Sines formula:
sin B = sin a * sin C / sin θ2
where a is the altitude, C is the difference in longitudes, and θ2 is the final latitude.

Since we are calculating the course on arrival, a will be the altitude, C will be the difference in longitudes, and θ2 will be the final latitude.

Hence, sin B = sin a * sin 4.53786 / sin θ2

2. Calculate the value of B by taking the inverse sine (arcsin) of that value:
B = arcsin(sin B)

This will give you the course on arrival (B).

By following these steps, you can calculate the altitude (a), longitude (N), or course on arrival (B) using spherical trigonometry for the given coordinates and distance travelled.

To solve this problem using spherical trigonometry, we can apply the laws of cosine and sine.

A) To calculate the altitude, we can use the law of cosine. The law of cosine states that in a spherical triangle, the cosine of one of the angles is equal to the sum of the cosines of the other two angles multiplied by the cosine of the opposite side.

In this case, let's define the sides and angles of the spherical triangle:
Angle A: 37° 45'N (latitude)
Angle B: 90° (right angle, since we are at the North Pole)
Angle C: 90° - 37° 45' = 52° 15'S (complementary angle to A)
Side a: Distance traveled = 2250 nm

We want to calculate side a (altitude) based on angles A and C:
Cos(A) = cos(C) * cos(a) + sin(C) * sin(a) * cos(B)

Substituting the values:
cos(37° 45'N) = cos(52° 15'S) * cos(a) + sin(52° 15'S) * sin(a) * cos(90°)

Now we can solve for cos(a) and find the altitude.

B) To calculate the longitude (N), we can use the law of sine. The law of sine states that in a spherical triangle, the sine of an angle is proportional to the sine of the opposite side.

In this case, we know:
Angle A: 37° 45'N (latitude)
Side a: Distance traveled = 2250 nm
Angle C: 90° - 37° 45' = 52° 15'S (complementary angle to A)
Side c: Longitude (N)

We want to calculate side c (longitude) based on angles A and C:
sin(C) / sin(c) = sin(A) / sin(a)

Substituting the known values, we can solve for sin(c) and find the longitude.

C) To calculate the course on arrival (B), we can also use the law of sine. Using the same logic as in part B, we can rearrange the equation to solve for angle B:

sin(B) / sin(b) = sin(A) / sin(a)

By substituting the known values, we can solve for sin(B) and find the course on arrival.