Question

Find the angle between a cube's diagonal and one of its edges.

Short answer

The angle between a cube's diagonal and one of its edges is approximately 54.74 degrees.

Step-by-step solution

Identify the right triangle

In a cube, a right triangle can be formed by one edge of the cube, the height from the edge to a vertex on the opposite face (which is also the length of a side since all edges of a cube are equal), and the diagonal connecting these two vertices.

Apply the concept of the trigonometric cosine function

The cosine of an angle in a right triangle is equal to the length of the adjacent side divided by the length of the hypotenuse. In this case, the length of the adjecent side is the edge of the cube, and the hypotenuse is the diagonal of the cube.

Calculate the length of the diagonal

By using Pythagoras theorem, you can find the length of the diagonal. Because a cube has all sides of equal length, if we assign \( a \) as the length of an edge, then the length of the diagonal \( d \) can be calculated as \( d = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3} \).

Apply the values to the cosine function

The cosine of the angle \( \theta \) between the diagonal and the edge of the cube is given by \( \cos(\theta) = \frac{a}{a\sqrt{3}} = \frac{1}{\sqrt{3}} \).

Find the angle

Finally, calculate the angle \( \theta \) by using the arccosine function to \( \frac{1}{\sqrt{3}} \). This gives an angle of approximately 54.74 degrees.