Law of sines — Safekipedia Discoverer

The law of sines is a key idea in trigonometry that helps us understand the relationship between the sides and angles of any triangle. It tells us that the ratio of a side's length to the sine of its opposite angle is the same for all three sides of the triangle. This means that if you know the length of one side and the measures of two angles, you can find the lengths of the other sides.

This rule is very useful in real life. For example, it can help mapmakers measure distances between places or allow engineers to design structures by calculating unknown measurements. The law of sines works for any triangle that is not a right triangle, and it is often used together with another rule called the law of cosines to solve more complex problems.

The law of sines can also help us understand shapes in more complicated spaces, but its most common use is in flat, two-dimensional geometry. It is one of the most important tools in trigonometry, a branch of math that studies angles and their relationships to sides in triangles.

Proof

To understand the law of sines, imagine a triangle with sides of different lengths. If we drop a line from one vertex to the opposite side, we can find relationships between the sides and the angles. By comparing these relationships, we see that the ratio of a side to the sine of its opposite angle stays the same for all three sides. This special number is connected to the size of the circle that can fit perfectly around the triangle.

Ambiguous case of triangle solution

When using the law of sines to find a missing side of a triangle, sometimes there can be two different triangles that fit the same information. This is called the ambiguous case. It happens when we know one angle and two sides of the triangle, but not enough to tell if there is one or two possible triangles.

For this to happen, the angle we know must be acute, meaning it is less than 90 degrees. If these conditions are met, there can be two different answers for the other angles and sides. This means the triangle is not fully determined until more information is known.

Examples

Here are some examples of how to use the law of sines to solve problems.

In the first example, we know one side of a triangle is 20 units, another side is 24 units, and one angle is 40 degrees. We want to find the unknown angle. Using the law of sines, we can calculate that the missing angle is about 32.39 degrees.

In the second example, we have a triangle where two sides are the same length, called x, and we know the length of the third side and one angle. The law of sines helps us find the other angles and confirm relationships between the sides and angles.

Relation to the circumcircle

The law of sines connects the sides and angles of a triangle to the size of its circumcircle — the circle that passes through all three vertices of the triangle. In simple terms, for any triangle, the length of each side divided by the sine of its opposite angle gives the same value. This value is twice the radius of the triangle's circumcircle.

The idea dates back to the ancient astronomer Ptolemy. It shows how the shapes of triangles relate to circles in a beautiful way.

Spherical law of sines

The spherical law of sines helps us understand triangles drawn on the surface of a sphere. These triangles have sides that are arcs of great circles — the longest possible circles you can draw on a sphere.

Imagine a sphere with a radius of 1 unit. The sides of a triangle on this sphere are arcs of great circles, and their lengths (let’s call them a, b, and c) are also the angles (in radians) at the center of the sphere. The angles of the triangle itself (let’s call them A, B, and C) are between the planes of these great circles.

The spherical law of sines states that for any such triangle:

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

This means that the ratio of the sine of each angle to the sine of its opposite side arc is the same for all three sides of the triangle. For smaller triangles on a large sphere, this law closely matches the regular law of sines used for flat triangles.

Hyperbolic case

In hyperbolic geometry, where the curvature is −1, the law of sines changes to a special form involving hyperbolic sine functions. It looks like this: sin A divided by sinh a equals sin B divided by sinh b equals sin C divided by sinh c.

When one of the angles, B, is a right angle, the formula simplifies to sin C equals sinh c divided by sinh b. This is similar to a well-known rule in regular geometry, where the sine of an angle can be expressed as the length of the opposite side divided by the hypotenuse.

The case of surfaces of constant curvature

In this section, we explore a special version of the law of sines that works in spaces where the shape is not flat. We use a special kind of sine function that changes based on a number called κ (kappa). This helps us understand triangles in curved spaces.

When we use this special sine function, the law of sines changes a little. It tells us how the sides and angles of a triangle relate in these curved spaces. This idea was discovered by a mathematician named János Bolyai.

Higher dimensions

A tetrahedron has four triangular facets. In higher dimensions, shapes like triangles and tetrahedrons are called n-dimensional simplexes. For these shapes, a special math rule works similarly to the law of sines, connecting the size of each part of the shape to angles and areas.

This rule shows that certain values, involving angles and areas of the shape’s parts, stay the same no matter which point you choose to look from. This helps us understand the shape’s size and how its parts relate to each other.

History

The idea behind the law of sines was known to the ancient astronomer Ptolemy and used in his work Almagest. Later, Indian mathematician Brahmagupta described related concepts in his books Brāhmasphuṭasiddhānta and Khaṇḍakhādyaka.

In the 10th century, Persian scholars began to work with the spherical version of the law. By the 13th century, Naṣīr al-Dīn al-Ṭūsī clearly stated and proved the law of sines for flat triangles, showing how it could help solve many triangle problems. Later, German mathematician Regiomontanus used the law as a foundation for solving triangles.