If a line makes angles of measure frac{pi}{6} | vedclass

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(B) Let the direction angles of the line be $\alpha = \frac{\pi}{6}$,$\beta = \frac{\pi}{3}$,and $\gamma$ be the angle with the $Z$ axis.The sum of th

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$\frac{\pi}{2}$

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(B) Let the direction angles of the line be $\alpha = \frac{\pi}{6}$,$\beta = \frac{\pi}{3}$,and $\gamma$ be the angle with the $Z$ axis.The sum of the squares of the direction cosines is always $1$,i.e.,$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the given values: $\cos^2(\frac{\pi}{6}) + \cos^2(\frac{\pi}{3}) + \cos^2 \gamma = 1$.We know that $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3}) = \frac{1}{2}$.So,$(\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2 + \cos^2 \gamma = 1$.$\frac{3}{4} + \frac{1}{4} + \cos^2 \gamma = 1$.$1 + \cos^2 \gamma = 1$.$\cos^2 \gamma = 0$,which implies $\cos \gamma = 0$.Therefore,$\gamma = \frac{\pi}{2}$.

If a line makes angles of measure $\frac{\pi}{6}$ and $\frac{\pi}{3}$ with $X$ and $Y$ axes respectively,then the angle made by the line with $Z$ axis is

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