A ventilation pipe runs through a factory in a straight line…

Step 1. Equal cosines. $l = m = n$ and $l^2 + m^2 + n^2 = 1$ give $3l^2 = 1$. So $l = \dfrac{1}{\sqrt{3}} \approx 0.577$ for the positive direction.

Step 2. The angle. $\alpha = \cos^{-1}\dfrac{1}{\sqrt{3}} \approx 54.74^\circ$, and the same for $\beta$ and $\gamma$.

Step 3. The sines. $\sin^2\alpha = 1 - \dfrac{1}{3} = \dfrac{2}{3}$. Three equal terms add to $3 \times \dfrac{2}{3} = 2$.