Question #73946

Let $\alpha = \sin^{-1}\left(\sin \frac{10\pi}{9}\right)$ and $\beta = \cos^{-1}\left(\cos \frac{10\pi}{9}\right)$. Let $A_1, A_2, \dots, A_8$ be $8$ arithmetic means inserted between $\alpha$ and $\beta$. Let $d$ be the common difference of this arithmetic progression, and define $k = \frac{\pi}{d}$. Consider three vectors $\vec{u}, \vec{v}, \vec{w}$ defined as: $\vec{u} = \sin^{-1}(\sin A_{k-5})\hat{i} + \cos^{-1}(\cos A_{k-3})\hat{j} + \tan^{-1}(\tan A_{k-1})\hat{k}$ $\vec{v} = \cos^{-1}(\cos A_{k-1})\hat{i} + \tan^{-1}(\tan A_{k-5})\hat{j} + \sin^{-1}(\sin A_{k-3})\hat{k}$ $\vec{w} = \tan^{-1}(\tan A_{k-3})\hat{i} + \sin^{-1}(\sin A_{k-1})\hat{j} + \cos^{-1}(\cos A_{k-5})\hat{k}$ Find the volume of the parallelepiped whose coterminous edges are represented by $\vec{u}, \vec{v}$, and $\vec{w}$.