You may be talking about something other than "[itex]\pi[/itex]". [itex]\pi[/itex] has a very specific value. There are a number of different ways to define [itex]\pi[/itex], one of them being the ratio between the circumference of a circle divided by the diameter of that circle in Euclidean Geometry. In a variety of forms of non-Euclidean geometries, that ratio might be something other than [itex]\pi[/itex] or the ratio might not be a constant. That has nothing to do with the number [itex]\pi[/itex].newjerseyrunner said:I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) = opposite / hypotenuse still hold true for all spaces?
In curved space, angle and arc length are not equivalent concepts. Unlike in Euclidean space where an angle is defined as the ratio of arc length to the radius of a circle, in curved space, the angle is determined by the curvature of the space itself. This means that two curves with the same arc length can have different angles in curved space.
In curved space, we use a mathematical concept called a geodesic to measure angles. Geodesics are the shortest paths between two points on a curved surface, and they follow the curvature of the space. The angle between two geodesics is the angle in curved space.
Yes, we can use trigonometric functions in curved space, but they are not the same as the traditional trigonometric functions used in Euclidean space. In curved space, the trigonometric functions are defined in terms of the geodesic distance between points, rather than the straight-line distance.
The curvature of space affects trigonometry by changing the relationships between angles and sides of a triangle. In Euclidean space, the sum of the angles in a triangle is always 180 degrees, but in curved space, the sum can be greater or less than 180 degrees depending on the curvature of the space. This means that trigonometric identities and formulas also change in curved space.
No, we cannot apply Euclidean trigonometry directly in curved space. Euclidean trigonometry is based on the assumption of a flat, two-dimensional space, while curved space has a non-zero curvature. This means that the formulas and concepts of Euclidean trigonometry do not hold true in curved space.