The angle between a diagonal of a cube and one of its edges is

Options

A. `cos^(-1)(1//sqrt(3))`
B. `pi//4`
C. `pi//6`
D. `pi//3`

Answer

To find the angle between a diagonal of a cube and one of its edges, we can follow these steps:

### Step 1: Understand the Geometry of the Cube
Consider a cube with edge length \( a \). The cube has vertices, edges, face diagonals, and a body diagonal.

### Step 2: Identify the Diagonal and Edge
- **Edge**: Any edge of the cube can be taken, say from point \( A \) to point \( B \).
- **Body Diagonal**: The diagonal that runs from one corner of the cube to the opposite corner, say from point \( A \) to point \( C \).

### Step 3: Calculate the Length of the Edge
The length of the edge is simply:
\[ \text{Length of edge} = a \]

### Step 4: Calculate the Length of the Face Diagonal
Using the Pythagorean theorem, the length of the face diagonal (which lies on one face of the cube) can be calculated as:
\[ \text{Length of face diagonal} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 5: Calculate the Length of the Body Diagonal
The body diagonal can also be calculated using the Pythagorean theorem:
\[ \text{Length of body diagonal} = \sqrt{a^2 + (a\sqrt{2})^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 6: Use Cosine to Find the Angle
To find the angle \( \theta \) between the body diagonal and the edge, we can use the cosine formula:
\[ \cos \theta = \frac{\text{Length of edge}}{\text{Length of body diagonal}} = \frac{a}{a\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Step 7: Calculate the Angle
To find \( \theta \), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

### Conclusion
Thus, the angle between a diagonal of a cube and one of its edges is:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

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