Hi,

In 2D, we can express the sides of a right angle triangle with sines and cosines. Is there a way to extend this idea to 3D? I.E. i have a right angle triangle with sides A and B, with magnitude of \(\sqrt(A^2+B^2)\) and angle \(\theta\).

Is there a similar analysis that can be applied to something that has a 3rd dimension? i.e. with magnitude of \(\sqrt(A^2+B^2+C^2)\)

 

Yep. And probably even if it has a 4th dimension.

 

Triangles are by definition 2D. You are wanting to deal with a pyramid if you use 3D. Could you draw what you are trying to describe?

 

well essentially i have something with magnitude \(\sqrt(A^2+B^2+C^2\). I want do be able to break this down into terms of sine and cosine, which usually only applies to 2d right angle triangles.

For example, for \(\sqrt(A^2+B^2)\) I can easily form a triangle with hypotenuse of that \(\sqrt(A^2+B^2)\) and sides with length A and B. Then i can use sin and cosine to describe \(A/(\sqrt(A^2+B^2))\) or \(B/\sqrt(A^2+B^2)\) quite easily. However, I am wondering if there is a similar approach or method that can be applied to geometries with magnitudes of \(\sqrt(A^2+B^2+C^2)\)

hope that makes my question more clear

 

If you have a rectangular box with sides of length A, B, and C, then the length of the diagonal is \(\sqrt{A^2+B^2+C^2}\). You should try to derive this for yourself. Start with any two sides (which line in a plane) and find the hypotenuse. This is the diagonal of that side. Now use that hypotenuse and the thrid side (which again define a plane) and find the hypotenuse of that. This IS the diagonal of the rectangular solid.