Question

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

• A. \(tan^{-1}(\frac{1}{\sqrt{2}})\)
• B. \(cos^{-1}(\frac{1}{3})\)
• C. \(cos^{-1}(\frac{1}{2\sqrt{2}})\)
• D. \(tan^{-1}(\sqrt{2})\)

Hint:

Take the cube with edge 1, diagonal \((1,1,1)\) and edge \((1,0,0)\).

Answer 1

The Correct Option is D

Step 1: Use a right triangle
Edge $=1$, face diagonal $=\sqrt2$, space diagonal $=\sqrt3$. The edge and face diagonal are perpendicular.

Step 2: Read the tangent
In the right triangle with the edge adjacent to $\theta$ and the face diagonal opposite, $\tan\theta=\sqrt2/1$. Option (D).

Final Answer:
The angle is $\tan^{-1}\sqrt2$, option (D). \[ \boxed{\tan^{-1}\sqrt2} \]