Find the angle between the diagonal of a cube and the diagonal of one of its faces.

Solution:

Let us consider a cube with side length = a.

Length of diagonal of cube on one of its faces = a√2

Length of diagonal of cube = a√3

To find the angle between the, use law of cosines

The law of cosines is given by

⇒ c² = a² + b² - 2ab cos C

Where c = length of side c, b = length of side b, a = length of side a

C = angle opposite to c

⇒ c² = (a√2)² + (a√3)² - 2 (a√2)(a√3) cos C

On simplification,

⇒ cosC = 2a²/(a√2 × a√3)

= 2/(√2 × √3)

= √2/√3

C = cos^-1(√2/√3)

Therefore, the angle between the diagonals is cos^-1(√2/√3).

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Find the angle between the diagonal of a cube and the diagonal of one of its faces.

Summary:

The angle between the diagonal of a cube and the diagonal of one of its faces is cos^-1(√2/√3).