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(A) Let the direction cosines of the line be $l, m, n$. Given that the line makes an angle $\alpha = \frac{\pi}{3}$ with both $X$ and $Y$-axes,we have

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$\frac{\pi}{4}$

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(A) Let the direction cosines of the line be $l, m, n$. Given that the line makes an angle $\alpha = \frac{\pi}{3}$ with both $X$ and $Y$-axes,we have $l = \cos(\frac{\pi}{3}) = \frac{1}{2}$ and $m = \cos(\frac{\pi}{3}) = \frac{1}{2}$.We know that $l^2 + m^2 + n^2 = 1$.Substituting the values: $(\frac{1}{2})^2 + (\frac{1}{2})^2 + n^2 = 1$.$\frac{1}{4} + \frac{1}{4} + n^2 = 1$ $\Rightarrow \frac{1}{2} + n^2 = 1$ $\Rightarrow n^2 = \frac{1}{2}$.Thus,$n = \pm \frac{1}{\sqrt{2}}$.For the acute angle $\gamma$ with the $Z$-axis,$\cos \gamma = |n| = \frac{1}{\sqrt{2}}$.Therefore,$\gamma = \cos^{-1}(\frac{1}{\sqrt{2}}) = \frac{\pi}{4}$.

If a line makes an angle of $\frac{\pi}{3}$ with each of the $X$ and $Y$-axes,then the acute angle made by the line with the $Z$-axis is:

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