A line makes equal angles with the diagonals of a cube. What is the sine of that angle?
- $\sqrt{\frac{2}{3}}$
- $\sqrt{\frac{1}{3}}$
- $\sqrt{\frac{1}{2}}$
-
none of these
The angle between a line and the four diagonals of a cube has a cosine value of 1/sqrt(3). The sine of that angle is sqrt(1 - cos^2) = sqrt(1 - 1/3) = sqrt(2/3).
Assume a cube with edge length 1 and position it in a three-dimensional coordinate system with vertices at (0,0,0) and (1,1,1). The four space diagonals can be represented by the direction vectors (1,1,1), (1,1,-1), (1,-1,1), and (-1,1,1). Let a line make equal angles with these diagonals, meaning its unit direction vector has equal magnitude dot products with all four vectors. By symmetry, we test the vector (1, 0, 0), which represents an edge of the cube; the dot product with each diagonal vector has a magnitude of 1. Because the magnitude of each diagonal vector is sqrt(3), the cosine of the angle between the edge and the diagonals is 1/sqrt(3). Using the identity sin^2 = 1 - cos^2, we find sin^2 = 1 - 1/3 = 2/3, making the sine of the angle sqrt(2/3).