Spherical trigonometry is a branch of geometry that deals with the study of spherical triangles, which are triangles drawn on the surface of a sphere. in this article, we have covered the definition of Spherical Trigonometry, some basic concepts related to the same and others in detail.

What is Spherical Trigonometry?

The study of the relationships between the sides and angles of triangles drawn on a sphere’s surface is known as spherical trigonometry. By using trigonometric concepts in non-planar geometry, it deals with the measurement and computation of angles, distances, and areas on spherical surfaces.

Spherical Trigonometry

Plane trigonometry, which deals with triangles on flat surfaces, is not the same as this field. The “sides” of triangles in spherical trigonometry are arcs of great circles on the surface of the sphere, and the angles between these arcs are determined at the spots where they cross.

For large-scale distance and direction computations on roughly spherical bodies like the Earth, as well as for astronomical computations and navigation, spherical trigonometry is especially crucial.

Basic Concepts of Spherical Trigonometry

Some basic concepts of Spherical Trigonometry include:

Spherical Triangle

A spherical triangle is a shape created on a sphere’s surface by three great circle arcs meeting at each of its three vertices pairwise. A few important facts regarding spherical triangles are:

• The sides are not lines; rather, they are the arcs of huge circles.
• Every side is quantified by its arc length, which is commonly represented as an angle at the sphere’s centre.
• A spherical triangle’s total angles are always larger than 180° and less than 540°.
• Any two sides added together always equal more than the third side.

Spherical Excess

A key idea in spherical trigonometry is spherical excess. It is the amount that a spherical triangle’s three angles added together surpass 180°.

Important spherical excess features are:

• Spherical excess E of a spherical triangle with angles A, B, and C is equal to E = (A + B + C) – 180°.
• Area of the spherical triangle is directly correlated with the spherical excess.
• Both the triangle’s size and the sphere’s radius affect how much of a spherical excess there is.

Formulas and Theorems Related to Spherical Trigonometry

Various formulas and theorems related to Spherical Trigonometry are:

Spherical Law of Cosines

This law relates the cosine of one side of a spherical triangle to the cosines of the other two sides and the sine of those sides times the cosine of the included angle.

For a spherical triangle with sides a, b, and c, and angles A, B, and C opposite these sides respectively, the law states:

cos(a) = cos(b)cos(c) + sin(b)sin(c)cos(A)

This formula can be written in two other equivalent forms by cyclic permutation of the sides and angles:

• cos(b) = cos(c)cos(a) + sin(c)sin(a)cos(B)
• cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)

There is also an alternative form relating an angle to three sides:

• cos(A) = -cos(B)cos(C) + sin(B)sin(C)cos(a)

Spherical Law of Sines

This law states that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.

For a spherical triangle with sides a, b, and c, and opposite angles A, B, and C, the law states:

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

Applications of Spherical Trigonometry

Various application of Spherical Trigonometry are in:

Navigation

• Maritime navigation: Calculating great circle routes for ships
• Aviation: Determining flight paths and distances between airports
• GPS systems: Computing positions and distances on Earth’s surface

Astronomy

• Calculating positions of celestial bodies
• Predicting eclipses and planetary movements
• Determining star rise and set times
• Mapping constellations

Geodesy and Cartography

• Measuring and mapping Earth’s surface
• Creating accurate projections of the globe onto flat maps
• Surveying large areas of land or sea

Meteorology

• Tracking the paths of storms and hurricanes
• Analyzing global wind patterns
• Studying the distribution of climate zones

Space Exploration

• Planning satellite orbits
• Calculating trajectories for space missions
• Determining optimal launch windows

Time Calculation

• Computing sunrise and sunset times
• Developing and adjusting calendars

Apart form those, there are various applications of Spherical Trigonometry

Examples on Spherical Trigonometry

Example 1: In a spherical triangle ABC, if a = 60°, b = 75°, and C = 90°, find angle A.

Solution:

Using the cosine formula for angles:

cos A = (cos a – cos b cos c) / (sin b sin c)

cos A = (cos 60° – cos 75° cos 90°) / (sin 75° sin 90°)

cos A = (0.5 – 0.2588 × 0) / (0.9659 × 1)

cos A = 0.5176

A ≈ 58.86°

Example 2: In a spherical triangle, if a = 30°, b = 45°, and c = 60°, find angle A.

Solution:

Using the cosine formula for angles:

cos A = (cos a – cos b cos c) / (sin b sin c)

cos A = (cos 30° – cos 45° cos 60°) / (sin 45° sin 60°)

cos A = (0.8660 – 0.7071 × 0.5) / (0.7071 × 0.8660)

cos A = 0.7500

A ≈ 41.41°

Example 3: In a spherical triangle ABC, if A = 60°, B = 90°, and c = 45°, find side a.

Solution:

Using the sine formula:

sin a / sin A = sin c / sin C

sin a = (sin c × sin A) / sin C

sin a = (sin 45° × sin 60°) / sin 90°

sin a = (0.7071 × 0.8660) / 1

sin a = 0.6124

a ≈ 37.76°

Example 4: In a spherical triangle, if a = 60°, b = 60°, and C = 60°, find angle A.

Solution:

Due to symmetry, A = B = 60°. This is an equilateral spherical triangle.

Question 5: In a spherical right triangle (C = 90°), if a = 30° and b = 45°, find angle A.

Solution 5: Using Napier’s rules for right spherical triangles:

tan A = tan a / sin b

tan A = tan 30° / sin 45°

tan A = 0.5774 / 0.7071

tan A = 0.8165

A ≈ 39.23°

Example 6: In a spherical triangle ABC, if A = 120°, B = 60°, and c = 90°, find side a.

Solution:

Using the cosine formula for sides:

cos a = cos b cos c + sin b sin c cos A

cos a = cos 90° cos 90° + sin 90° sin 90° cos 120°

cos a = 0 + 1 × (-0.5)

cos a = -0.5

a = arccos(-0.5) ≈ 120°

Example 7: In a spherical right triangle (C = 90°), if b = 60° and c = 45°, find angle A.

Solution:

Using Napier’s rules for right spherical triangles:

cos A = tan b cot c

cos A = tan 60° × cot 45°

cos A = 1.7321 × 1

cos A = 1.7321

A = arccos(1.7321) ≈ 0° (Note: This is not possible in a real spherical triangle, indicating an issue with the given values)

Example 7: In a spherical triangle ABC, if a = 80°, b = 70°, and C = 100°, find angle B.

Solution:

Using the sine formula:

sin B / sin b = sin C / sin c

sin B = (sin b × sin C) / sin c

First, we need to find c using the cosine formula:

cos c = cos a cos b + sin a sin b cos C

cos c = cos 80° cos 70° + sin 80° sin 70° cos 100°

cos c ≈ 0.1736 c ≈ 80.01°

Now we can solve for B:

sin B = (sin 70° × sin 100°) / sin 80.01°

sin B ≈ 0.9397

B ≈ 70.24°

Example 8: In a spherical triangle, if A = 60°, B = 75°, and c = 50°, find side a.

Solution:

Using the sine formula:

sin a / sin A = sin c / sin C

sin a = (sin A × sin c) / sin C

We need to find C first using the cosine formula for angles:

cos C = -cos A cos B + sin A sin B cos c

cos C = -cos 60° cos 75° + sin 60° sin 75° cos 50°

cos C ≈ 94.92°

Now we can solve for a:

sin a = (sin 60° × sin 50°) / sin 94.92°

a ≈ 41.52°

Real-Life Examples on Spherical Trigonometry

A real world examples of using Spherical Trigonometry is:

Example: Great Circle Navigation

Suppose a pilot needs to fly from New York City (40.7°N, 74.0°W) to Tokyo (35.7°N, 139.8°E). We want to find:

• The shortest distance between these cities (along a great circle)
• Initial bearing (direction) from New York to Tokyo

Step 1: Convert the Locations to Radians

New York: φ₁ = 40.7° × π/180 = 0.7101 radians,

λ₁ = -74.0° × π/180 = -1.2915 radians

Tokyo: φ₂ = 35.7° × π/180 = 0.6230 radians,

λ₂ = 139.8° × π/180 = 2.4400 radians

Step 2: Calculate the central angle (Δσ) using the spherical law of cosines

cos(Δσ) = sin(φ₁)sin(φ₂) + cos(φ₁)cos(φ₂)cos(λ₂ – λ₁)

Δσ = arccos{sin(0.7101)sin(0.6230) + cos(0.7101)cos(0.6230)cos(2.4400 – (-1.2915))}

Δσ ≈ 1.9853 radians

Step 3: Calculate Distance (d)

Assuming Earth’s radius (R) is 6371 km:

d = R × Δσ

d ≈ 6371 × 1.9853 ≈ 12,647 km

Step 4: Calculate Initial Bearing (θ)

θ = arctan(sin(λ₂-λ₁)cos(φ₂), cos(φ₁)sin(φ₂) – sin(φ₁)cos(φ₂)cos(λ₂-λ₁))

θ ≈ 0.5951 radians

Convert to degrees: 0.5951 × 180/π ≈ 34.1°

Results:

• Shortest distance between New York City and Tokyo is approximately 12,647 km.
• Initial bearing from New York to Tokyo is approximately 34.1° (measured clockwise from due North).

FAQs on Spherical Trigonometry

What is Spherical Trigonometry?

Spherical Trigonometry is the branch of mathematics that deals with the relationships between sides and angles of triangles on the surface of a sphere.

What is a Great Circle?

Great Circle is the intersection of a sphere with a plane passing through the sphere’s center, representing the shortest path between two points on a sphere.

How is Spherical Trigonometry used in Navigation?

Spherical Trigonometry is used to calculate distances, bearings, and routes for ships and aircraft traveling long distances on the Earth’s surface.

What is the Polar Triangle?

Polar Triangle is a triangle on a sphere formed by the poles of the great circles containing the sides of another spherical triangle.

Are there Right Angles in Spherical Triangles?

Yes, spherical right triangles exist and have special properties, often simplified using Napier’s rules.

How does Spherical Excess relate to Area of a Spherical Triangle?

Spherical excess (sum of angles minus 180°) is proportional to the area of the spherical triangle.

What are Real-World Applications of Spherical Trigonometry?

Some real-world applications of Spherical Trigonometry includes, navigation, astronomy, geodesy, cartography, and global positioning systems (GPS).

How does Shape of Earth Affect Calculations in Spherical Trigonometry?

While the Earth isn’t a perfect sphere, spherical trigonometry provides good approximations for many practical purposes. For highest precision, the Earth’s ellipsoidal shape is considered.