Answer
See explanations.
Work Step by Step
Step 1. Identify the given quantities: $\alpha=150^{\circ}$ and $\gamma=25^{\circ}$
Step 2. Recall the properties of the direction cosines $cos^2\alpha+cos^2\beta+cos^2\gamma=1$
Step 3. Test the given values for the above equation: $cos^2150^{\circ}+cos^225^{\circ}+cos^2\beta=1$ which gives $0.75+0.82+cos^2\beta=1$ or $cos^2\beta=-0.57$.
Step 4. Since $cos^2\beta\geq0$. the above result can not be true, which means that the given angles are impossible direction angles.