How to Solve Spherical Trigonometry Problems with PDF Download

Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

In this article, we will introduce some basic concepts and formulas of spherical trigonometry, and show you how to solve some common problems with PDF download of solved exercises.

What is a spherical triangle?

A spherical triangle is a figure formed by three great circles on the surface of a sphere. A great circle is a circle that divides the sphere into two equal hemispheres. The vertices of a spherical triangle are the points where the great circles intersect, and the sides are the arc lengths of the great circles between the vertices.

The angles of a spherical triangle are the dihedral angles between the planes that contain the great circles. The sum of the angles of a spherical triangle is always greater than 180 degrees, and less than 540 degrees. The difference between 180 degrees and the sum of the angles is called the spherical excess, and it is proportional to the area of the spherical triangle.

What are some basic formulas of spherical trigonometry?

There are many formulas that relate the sides and angles of a spherical triangle, but we will only mention some of the most important ones here. For more details, you can refer to Wikipedia or Wolfram MathWorld.

Let a spherical triangle have sides α, β, and γ, and angles A, B, and C opposite to them. Then we have:

The cosine rules for sides:
cos α = cos β cos γ + sin β sin γ cos A
cos β = cos γ cos α + sin γ sin α cos B
cos γ = cos α cos β + sin α sin β cos C
The sine rule for sides:
sin α / sin A = sin β / sin B = sin γ / sin C
The cosine rules for angles:
cos A = -cos B cos C + sin B sin C cos α
cos B = -cos C cos A + sin C sin A cos β
cos C = -cos A cos B + sin A sin B cos γ
The area formula:
Area = R^2 E
where R is the radius of the sphere, and E is the spherical excess.

How to solve spherical trigonometry problems?

To solve spherical trigonometry problems, we need to apply the appropriate formulas to find the unknown sides or angles of a given spherical triangle. Depending on the given information, we may need to use different methods or combinations of formulas.

Here are some examples of common types of problems:

Given three sides, find an angle.
In this case, we can use one of the cosine rules for sides to find an angle. For example, if we know α, β, and γ, we can find A by solving:
cos A = (cos α – cos β cos γ) / (sin β sin γ)
Given two sides and an angle opposite to one of them, find another angle.
In this case, we can use one of the cosine rules for angles to find another angle. For example, if we know α, β, and C, we can find A by solving:
cos A = (-cos C + cos α cos β) / (sin α sin β)
Given two angles and a side opposite to one of them, find another side.
In this case, we can use one of the sine rules for sides to find another side. For example, if we know A, B, and α, we can find β by solving:
sin β = (sin α / sin A) sin B
Given three angles, find a side.
In this case, we can use one of the cosine rules for sides or angles to find a side. For example, if we know A, B, and C, we can find α by solving:
cos α = -cos B cos C + sin B sin C cos A

Where can I find more exercises and solutions for spherical trigonometry?

If you want to practice your skills and test your knowledge of spherical trigonometry, you can find many exercises and solutions online. Here are some sources that you can use:

Spherical Trigonometry by Rob Johnson: This is a PDF document that contains a brief introduction to spherical trigonometry, followed by 24 exercises with solutions.
Spherical Trigonometry by I. Todhunter: This is a classic textbook that covers all the main topics and formulas of spherical trigonometry, with numerous examples and exercises. You can download it for free from Project Gutenberg.
Plane and Spherical Trigonometry by J.B. Tatum: This is a chapter from a book on celestial mechanics that explains some applications of spherical trigonometry to astronomy, with 10 exercises and solutions.

These sources will help you learn more about spherical trigonometry and improve your problem-solving skills. You can also search for more exercises and solutions on the internet, or create your own problems using the formulas and methods that you have learned.

What are some tips or tricks for solving spherical trigonometry problems?

Solving spherical trigonometry problems can be challenging, especially if you are not familiar with the formulas and methods. Here are some tips or tricks that may help you:

Draw a clear and accurate diagram of the spherical triangle, and label all the given and unknown sides and angles. This will help you visualize the problem and choose the appropriate formulas.
Use the mnemonic ABC to remember the order of the sides and angles of a spherical triangle: A is opposite to α, B is opposite to β, and C is opposite to γ.
Use the mnemonic SAS to remember the cosine rules for sides: Side-Angle-Side. For example, to find the side opposite to A, use the formula that involves the sides adjacent to A and the angle between them.
Use the mnemonic SSS to remember the sine rule for sides: Side-Side-Side. For example, to find the ratio of two sides, use the formula that involves the sines of the opposite angles.
Use the mnemonic ASA to remember the cosine rules for angles: Angle-Side-Angle. For example, to find the angle opposite to α, use the formula that involves the angles adjacent to α and the side between them.
Use a calculator or a table of trigonometric functions to find the values of sines and cosines. Make sure you set your calculator to radians mode, and round your answers to a suitable degree of accuracy.
Check your answers by substituting them back into the original problem or using another formula. If your answers are correct, they should satisfy all the given conditions and constraints.

These tips or tricks will help you solve spherical trigonometry problems more easily and efficiently. You can also practice more exercises and solutions to gain more confidence and experience.