The usage of the Law of Cosine and the Law of Sine of spherical trigonometry Essay

To check the distortions of the shortest distance between two points, an equation can be derived from the Law of Cosine and the Law of Sine of spherical trigonometry that finds the great circle distance between two points on a sphere. In the diagrams above, a, b and c are the angles in degrees on a sphere as can be seen from Fig 1.4. They are the angles subtending the arcs BC, AC and AB respectively. A, B and C refer to the different points on Earth but will also be used to refer to the angles between the arcs at those points. The points A, B and C are identified using the longitudes and latitudes shown in the diagrams.

Referring to the diagram shown above

Law of Cosine for spherical triangles cos(c)=cos(b) cos(a)+sin(b) sin(a)cos(C)

Law of Sine for spherical triangles (sin(A))/(sin(a))=(sin(B))/(sin(b))=(sin(C))/(sin(c))

The derivation of the above laws is not within the scope of this exploration. The above laws cannot be used as it is because when we talk about Earth, we use longitudes and latitudes so the appropriate formulae need to be derived from the above.

From Fig 1.4, we can see that a=902 and b=901 with 2 and 1 being the latitudes of the points B and A respectively. Since 2 and 1 are the respective longitudes, C= 2 1 since C is the angle between the arcs a and b. Therefore,

Since, cos(c)=cos(b) cos(a)+sin(b) sin(a)cos(C)

cos(c)=cos(90_1 ) cos(90_2 )+sin(90_1 ) sin(90_2 )cos(_2_1)

cos(c)=sin(_1 ) sin(_2 )+cos(_1 ) cos(_2 ) cos(_2_1 )

c=|arccos(sin(_1 ) sin(_2 )+cos(_1 ) cos(_2 ) cos(_2_1 ))|

The distance, s, between the two locations, in this case A and B, can then be calculated from their longitudes and latitudes since s=R. R is the average radius of Earth (6371km) and =c.

s=R|arccos(sin(_1 ) sin(_2 )+cos(_1 ) cos(_2 ) cos(_2_1 ) ) |

Note For the above formula, the longitudes and latitudes have to be converted to radians.

However, this formula assumes that the Earth is a perfect sphere, which it is not. The Earth is more of an ellipsoid and even then, it is not a perfect ellipse and thus making perfectly accurate formulae and representations of Earth is almost impossible. However, from a pilots perspective, this does not make a huge difference. A pilot will care about distance travelled for a few reasons  time and money. The amount of time taken to travel a shorter distance will clearly be less than the time taken to travel a longer distance. However, the values obtained from using this formula varies from the true value minimally. Furthermore, to a pilot, travelling a greater distance would change the fuel considerations of the aircraft but again, the difference between the true value and the values obtained from this formula is too minute for it to make a difference. However this distortion will change the further the distances get, as there is less likelihood of the line connecting the two locations to be on a perfect sphere.