Cyclic cubings

Geometry Level 5

∑ cyc ( a , b , c , d ) cos ⁡ 2 ( a ) \large\displaystyle\sum_{\text{cyc}(a,b,c,d)}\cos^{2}(a)

A line makes angles a , b , c , d a,b,c,d with four diagonals of a cube. Evaluate the summation above.


The answer is 1.333.

Shivam Jadhav by Shivam Jadhav , 22, India

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2 solutions

Ashutosh Sharma
Jan 28, 2018

assume a line through centre of cube paralle to 4 sides ,thus angle betwwen line and diagonal is arctan(sqrt2) thus 4 times cos^2(arctan(sqrt2) give 4/3

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Aaaaa Bbbbb
Aug 8, 2015

Assume that this line is one of the diagonal. It is very easy to calculate the angles of this line with four diagonals: 0 0 , 7 0 0 , 7 0 0 , 7 0 0 0^{0}, 70^{0}, 70^{0}, 70^{0} So: S = 1.333 S=\boxed{1.333}

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