Angle and trig definitions in curved space

[newjerseyrunner]

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?

 

[robphy]

Just to clarify... $\pi=4\arctan(1)=3.1415...$
It's a particularly interesting number, which is also equal to the circumference/diameter ratio of any circle on a Euclidean plane.
Now in a curved space, the circumference/diameter ratio is no longer independent of the circle... and many formulas that work for (say) triangles in the plane don't work any more. [You must do calculus and differential geometry now.]
However, at any point in a curved space, there is a tangent vector space there. On a Euclidean plane in that vector space, that ratio is still $\pi$ for any circle drawn on it.

 

[HallsofIvy]

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?

You may be talking about something other than "$\pi$". $\pi$ has a very specific value. There are a number of different ways to define $\pi$, 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 $\pi$ or the ratio might not be a constant. That has nothing to do with the number $\pi$.