Final answer:
The angle between a diagonal of a cube and a diagonal of one of its faces is found using trigonometric relationships, involving the Pythagorean theorem and the cosine formula. After calculations, the angle is determined by taking the inverse cosine of the ratio of a face diagonal to a cube diagonal.
Step-by-step explanation:
To find the angle between a diagonal of a cube and a diagonal of one of its faces, we apply some trigonometric relationships. The diagonal of the cube can be seen as the hypotenuse of a right-angled triangle where the sides of this triangle are a face diagonal and an edge of the cube. We denote the length of the edge of the cube as a. Therefore, the length of the face diagonal, using the Pythagorean theorem, is a√2, and the length of the cube diagonal is a√3. The angle between the cube diagonal and the face diagonal can be calculated using the cosine formula:
cos(θ) = adjacent side / hypotenuse = (a√2) / (a√3) = √2/√3 = √(2/3)
To find the angle θ, we take the inverse cosine (arccos) of √(2/3):
θ = arccos(√(2/3))
Rounded to one decimal place, we obtain the angle θ.