# Law of Sines (Elliptic, Hyperbolic, Euclidean)

• DarthPickley
In summary, the conversation discusses the Law of Sines and its applications in different geometries such as the Euclidean Plane, Sphere, and Hyperbolic Plane. The Law of Sines states that for a triangle with angles A, B, and C and side lengths a, b, and c, the ratio of any side to the sine of its opposite angle is constant. The conversation also mentions proofs for the Euclidean and Spherical laws of cosines and the difficulty in proving the Spherical and Hyperbolic laws of sines. It is suggested to look into the use of geometric algebra to handle different geometries.
DarthPickley
Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries.

http://en.wikipedia.org/wiki/Law_of_sines
http://mathworld.wolfram.com/LawofSines.html

The Law of Sines states that, for a triangle ABC with angles A, B, C, and side lengths a = BC, b = AC, & c = AB, which is in:
The Euclidean Plane:
a/Sin(A) = b/Sin(B) = c/Sin(C)​
The Sphere:
Sin(a)/Sin(A) = Sin(b)/Sin(B) = Sin(c)/Sin(C)​
The Hyperbolic Plane:
Sinh(a)/Sin(A) = Sinh(b)/Sin(B) = Sinh(c)/Sin(C)​

I also know that for Euclidean geometry, a/Sin(A) is the radius of the circumscribed circle.

Here is a proof for Euclidean geometry:
Given Triangle ABC:
Construct the altitude from C. Let h be the length of this altitude (the height, where AB is the base)
By a definition of Sine, Sin(A) = h/b and similarly, Sin(B) = h/a.
h = b*Sin(A) and h = a*Sin(B)
b*Sin(A) = h = a*Sin(B) ==> a*Sin(B) = b*Sin(A)
a/Sin(A) = b/Sin(B)
since A and B can be chosen to be any two vertices of the same triangle, it is also true that a/Sin(A) = c/Sin(C) = b/Sin(B) : which is that which was to be demonstrated.

Now, I have been able to prove the Spherical law of cosines using the Sphere and some basic Linear Algebra like dot product = cosine of angle. I don't really understand the hyperbolic plane well, but the formula is similar for hyperbolic law of cosines.

I have been less successful proving the Spherical law of sines, not to mention Hyperbolic law of sines.
This is what I am asking for help with. It may require some stuff with vectors I don't understand right now, but if someone can explain it in a way that makes more sense.

Is there a way to say the theorems I used for the Euclidean proof that is for the other geometries? Of course, you can derive from the law of sines itself that if C is a right angle, then Sin(A) = Sinh(a)/Sinh(c). This is sufficient to prove the entire Law of Sines. But is there a way to prove these Right-Triangle Trigonometric Definitions for Sine / Sinh for Spherical and Hyperbolic Geometry? Is this easier or simpler than the whole thing?

http://mathworld.wolfram.com/SphericalTrigonometry.html

I don't have any experience with working out the details, but I know that papers and textbooks about geometric algebra talk about how you can treat Euclidean, spherical, and hyperbolic geometries in a unified way. Off the top of my head: try the chapter in Lasenby and Doran's "Geometric Algebra for Physicists" dealing with different geometries -- I think it's chapter 10 or 11. That's where I would start.

The conformal model for geometric algebra, which models points in R(0,n) with vectors in R(1,n+1) is the tool that's used. It seems very powerful; I'm hoping to learn it in more detail so I can understand how it's used to handle the crystallographic space groups.

## 1. What is the Law of Sines?

The Law of Sines is a mathematical rule that relates the sides and angles of a triangle in a non-Euclidean space. It states that the ratio of the sine of an angle to the length of its opposite side is constant for all three angles in a triangle.

## 2. What are the three types of non-Euclidean spaces?

The three types of non-Euclidean spaces are elliptic, hyperbolic, and Euclidean. Each type has its own set of geometric rules and properties that differ from the Euclidean space, which is the traditional geometry we learn in school.

## 3. How does the Law of Sines apply to elliptic geometry?

In elliptic geometry, the Law of Sines is used to find the sides and angles of a triangle on a spherical surface. The angles in this type of geometry are measured in radians, and the Law of Sines is modified to account for the curvature of the surface.

## 4. Can the Law of Sines be used in hyperbolic geometry?

Yes, the Law of Sines can also be applied in hyperbolic geometry, which is a type of geometry that has a constant negative curvature. However, the formula is modified to account for the negative curvature of the space.

## 5. What are some real-world applications of the Law of Sines?

The Law of Sines has many practical applications, such as in navigation to find the distance and direction between two points on a sphere, in astronomy to determine the position of celestial bodies, and in surveying to measure distances and angles on the Earth's surface.

• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Precalculus Mathematics Homework Help
Replies
8
Views
5K
• General Math
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
14
Views
539
• Precalculus Mathematics Homework Help
Replies
5
Views
1K
• Introductory Physics Homework Help
Replies
12
Views
2K
• Precalculus Mathematics Homework Help
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
2K