Exercise 2 – The application of spherical trigonometry in the solution of navigational problems.
This post continues the series of navigation related exercises which were requested by navigation and maths teachers. When the post expires, its contents will be included under the menu heading ‘Navigation Exercises’ on the home page.
Objective. To exercise the application of the cosine rule in the solution of spherical triangles.
Part 1. Revising the application the cosine rule in the solutions of plane triangles.
Reminder. The cosine rule states that a2 = b2 + c2 – 2bc Cos.(A)
a2 = b2 + c2 – 2bc Cos.(A)
substituting the above values gives: a2 = 73 – 48 Cos.(32o)
= 32.3 cm
therefore, a = 5.68 cm
Part 2. Applying the cosine rule in the solution of a spherical triangle.
To revise spherical trigonometry click here.
As explained in the link above, we amend the cosine rule when we are working with spherical triangles. For example, to find side a in the spherical triangle shown in the diagram below we have: Cos(a) = [Cos(b) . Cos(c)] +[Sin(b) . Sin(c) .Cos(A)]
Example. Find side a given that b = 7.5m., c = 5.9m., A = 49o
Cos(a) = [Cos(b) . Cos(c)] +[Sin(b) . Sin(c) .Cos(A)]
Substituting the values given above, we have:
Cos(a) = [Cos(7.5).Cos(5.9)] + [Sin(7.5). Sin(5.9).Cos(49o)]
= [0.991 x 0.995] + [0.13 x 0.10 x 0.66]
= 0.986 + 0.009
therefore a = 5.73m.
Part 3. Applying spherical trigonometry in the solution of a navigational problem.
In the diagram above,
Angle ZPX represents the time-angle,
P represents the North Pole,
Z represents the position of an observer on the Earth’s surface,
X represents the geographical position of the Sun (GP),
XZ represents the angular distance of the observer from the GP of the Sun,
EX represents the declination of the Sun,
PX represents 90o – the declination,
CZ represents the latitude of the observer,
PZ represents 90o – latitude,
E represents a point on the Equator that lies on the same meridian of longitude as point X,
C represents a point on the Equator that lies on the same meridian of longitude as point Z.
Task: Given the following: Lat. = 45o North, Dec. = 18o North, and Time = 3 hours after noon, find the distance of the observer from the Sun’s GP.
EX = 18o therefore PX = 90o – 18o = 72o
CZ = 45o therefore PZ = 90o – 45o = 45o
Angle ZPX = 45o (since 3 hours = a time angle of 3 x 15o).
Reminder. As stated above, the formula for finding the third side of a spherical triangle, when the other two sides and their included angle is known, is:
Cos.(a) = [Cos.(b) . Cos.(c)] + [Sin(b) . Sin(c) .Cos.(A)]
Applying this formula to the diagram above, we have:
Cos.(XZ) = [Cos.(PZ) x Cos.(PX)] + [Sin(PZ) x Sin(PX) x Cos.(ZPX)]
Substituting the values above gives:
Cos.(XZ) = [Cos.(45o) x Cos.(72o)] + [Sin (45o) x Sin(72o) x Cos.(45o)]
Cos.(XZ) = [0.707 x 0.309] + [0.707 x 0.951 x 0.707]
Cos.(ZX) = 0.218 + 0.475 = 0.693
therefore ZX = 46.13 = 2767.8′
therefore distance of observer from GP = 2767.8 n.m. (since 1′ of arc on the earth’s surface = approx. 1 nautical mile).
A fuller explanation of the application of spherical trigonometry in astro navigation is given in the book ‘Astro Navigation Demystified’.
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