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Let a cube be divided into three pieces by two parallel planes, each of which go through exactly three vertices. (Think of a cube with a vertex at the top, a vertex at the bottom and two horizontal planes) What fraction of the cube is above the top plane, between the planes and below the bottom plane (obviously the first and last answer are the same). I think this can be done by calculating the volume of the pyramid above the top plane with edge of 1 unit, the height of the pyramid is 1/3 the solid diagonal or sqrt(3)/3 and the base of the pyramid is a triangle made up of face diagonals, etc. Is this the way to do it, or is there a clean fast way (preferably one that could be extended to do the equivalent slices of a hypercube... Naraht ( talk) 02:34, 1 October 2016 (UTC)
Someone told me that anyone can calculate the values of Sine and Cosine in degrees using nothing but a single complex number.
Define a complex number value "onedeg" as
onedeg = 1 * (Cos[1 deg] + i Sin[1 deg]) = 0.999847695156391 + 0.0174524064372835 i
Next calculate Sine[23.45 deg] and Cos[23.45 deg]
onedeg^23.45 = 0.917408 + 0.397949 i
voila! Cos[23.45 deg] = 0.917408 and Sin[23.45 deg] = 0.397949
It feels like magic, a single complex number is capable of calculating Sine and Cosine in degrees! 110.22.20.252 ( talk) 13:25, 1 October 2016 (UTC)
Mathematics desk | ||
---|---|---|
< September 30 | << Sep | October | Nov >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Let a cube be divided into three pieces by two parallel planes, each of which go through exactly three vertices. (Think of a cube with a vertex at the top, a vertex at the bottom and two horizontal planes) What fraction of the cube is above the top plane, between the planes and below the bottom plane (obviously the first and last answer are the same). I think this can be done by calculating the volume of the pyramid above the top plane with edge of 1 unit, the height of the pyramid is 1/3 the solid diagonal or sqrt(3)/3 and the base of the pyramid is a triangle made up of face diagonals, etc. Is this the way to do it, or is there a clean fast way (preferably one that could be extended to do the equivalent slices of a hypercube... Naraht ( talk) 02:34, 1 October 2016 (UTC)
Someone told me that anyone can calculate the values of Sine and Cosine in degrees using nothing but a single complex number.
Define a complex number value "onedeg" as
onedeg = 1 * (Cos[1 deg] + i Sin[1 deg]) = 0.999847695156391 + 0.0174524064372835 i
Next calculate Sine[23.45 deg] and Cos[23.45 deg]
onedeg^23.45 = 0.917408 + 0.397949 i
voila! Cos[23.45 deg] = 0.917408 and Sin[23.45 deg] = 0.397949
It feels like magic, a single complex number is capable of calculating Sine and Cosine in degrees! 110.22.20.252 ( talk) 13:25, 1 October 2016 (UTC)