Spherical Trigonometry

Spherical Trigonometry

The study of triangles on the surface of a sphere, the sides of which are arcs of great circles. Useful for navigation.

Sphere with spherical triangle ABC drawn on its surface, where vertices A and C sit on the equator and B at the top.

See also

Key Formula

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

Where:

$a$ = Angular measure (in radians or degrees) of the side opposite vertex A — the arc length divided by the sphere's radius
$b$ = Angular measure of the side opposite vertex B
$c$ = Angular measure of the side opposite vertex C
$C$ = The angle at vertex C, formed between the two great-circle arcs meeting at C

Worked Example

Problem: On a sphere, a triangle has sides a = 60°, b = 50°, and the included angle C = 90°. Find the angular length of side c using the spherical law of cosines.

Step 1: Write the spherical law of cosines for side c.

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

Step 2: Substitute the known values: a = 60°, b = 50°, C = 90°. Note that cos 90° = 0.

\cos c = \cos 60° \cos 50° + \sin 60° \sin 50° \cdot 0

Step 3: Since the second term vanishes, compute the first term.

\cos c = (0.5)(0.6428) = 0.3214

Step 4: Take the inverse cosine to find c.

c = \cos^{-1}(0.3214) \approx 71.25°

Answer: The angular length of side c is approximately 71.25°. On a sphere of radius R, the actual arc length would be R × 71.25° × (π/180).

Another Example

This example uses the law of cosines in reverse — finding an unknown angle from three known sides, rather than finding a side from two sides and an included angle.

Problem: A spherical triangle has sides a = 80°, b = 70°, and c = 60°. Find the angle A opposite side a using the spherical law of cosines for angles.

Step 1: Rearrange the spherical law of cosines to solve for cos A. The formula solved for an angle is:

\cos A = \frac{\cos a - \cos b \cos c}{\sin b \sin c}

Step 2: Substitute the values a = 80°, b = 70°, c = 60°.

\cos A = \frac{\cos 80° - \cos 70° \cos 60°}{\sin 70° \sin 60°}

Step 3: Evaluate numerator and denominator separately. cos 80° ≈ 0.1736, cos 70° ≈ 0.3420, cos 60° = 0.5, sin 70° ≈ 0.9397, sin 60° ≈ 0.8660.

\cos A = \frac{0.1736 - (0.3420)(0.5)}{(0.9397)(0.8660)} = \frac{0.1736 - 0.1710}{0.8138}

Step 4: Compute the result.

\cos A = \frac{0.0026}{0.8138} \approx 0.003195

Step 5: Take the inverse cosine.

A = \cos^{-1}(0.003195) \approx 89.82°

Answer: Angle A is approximately 89.82°, which is very close to a right angle.

Frequently Asked Questions

What is the difference between spherical trigonometry and plane trigonometry?

Plane trigonometry deals with triangles on a flat surface where angles always sum to exactly 180°. Spherical trigonometry deals with triangles on a sphere where angles sum to more than 180° (and can reach up to 540°). The formulas differ: the plane law of cosines is c² = a² + b² − 2ab cos C, while the spherical version uses cosines and sines of the sides themselves rather than their lengths squared.

When do you use spherical trigonometry?

You use spherical trigonometry whenever you need to measure distances or angles on a sphere or near-spherical surface. The most common applications are navigation (finding the shortest route between two points on Earth), astronomy (calculating positions of stars on the celestial sphere), and satellite communication (determining signal paths). Any problem involving great-circle distances on a globe calls for spherical trigonometry.

What is spherical excess?

Spherical excess is the amount by which the sum of the three angles of a spherical triangle exceeds 180°. It is defined as E = A + B + C − 180°. The area of a spherical triangle on a unit sphere equals its spherical excess measured in radians. For example, a triangle with angles 100°, 80°, and 70° has a spherical excess of 70°.

Spherical Trigonometry vs. Plane Trigonometry

	Spherical Trigonometry	Plane Trigonometry
Surface	Triangles on the surface of a sphere	Triangles on a flat plane
Sides	Arcs of great circles, measured as angles	Straight line segments, measured as lengths
Angle sum	Greater than 180° (up to 540°)	Exactly 180°
Law of cosines	cos c = cos a cos b + sin a sin b cos C	c² = a² + b² − 2ab cos C
Typical use	Navigation, astronomy, geodesy	Surveying, engineering, general geometry

Why It Matters

Spherical trigonometry appears whenever you work with distances on Earth's surface — airline routes, ship navigation, and GPS calculations all rely on it. In astronomy, it is used to convert between coordinate systems (equatorial, ecliptic, horizontal) on the celestial sphere. Students encounter it in advanced geometry, physics, and any course involving navigation or Earth sciences.

Common Mistakes

Mistake: Using the plane law of cosines (c² = a² + b² − 2ab cos C) on a spherical triangle.

Correction: On a sphere, sides are angular measures, not lengths. You must use the spherical law of cosines: cos c = cos a cos b + sin a sin b cos C. The plane formula only approximates correctly for very small triangles.

Mistake: Assuming the angles of a spherical triangle sum to 180°.

Correction: On a sphere, the angle sum is always greater than 180°. The excess above 180° is called the spherical excess and is directly related to the triangle's area. Forgetting this leads to incorrect angle calculations.

Related Terms

Trigonometry — The broader field that spherical trig extends
Triangle — The fundamental shape studied on the sphere
Sphere — The surface on which spherical triangles exist
Great Circle — Sides of a spherical triangle are great-circle arcs
Arc of a Circle — Each side is an arc measured by its central angle
Surface — Spherical trig operates on a curved surface
Side of a Polygon — Analogous concept for sides on a sphere