If a line makes angles frac{pi}{3} and frac{p | vedclass

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(D) Let the angles made by the line with the positive $x$,$y$,and $z$-axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = \frac{\pi}{3}

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$\frac{\pi}{3}$

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(D) Let the angles made by the line with the positive $x$,$y$,and $z$-axes be $\alpha$,$\beta$,and $\gamma$ respectively.Given $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{4}$.The direction cosines of the line are $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$.We know that $l^2 + m^2 + n^2 = 1$,which implies $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$.Substituting the values: $\cos^2(\frac{\pi}{3}) + \cos^2(\frac{\pi}{4}) + \cos^2 \gamma = 1$.$(\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 + \cos^2 \gamma = 1$.$\frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1$.$\frac{3}{4} + \cos^2 \gamma = 1$.$\cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4}$.$\cos \gamma = \frac{1}{2}$ (since the angle is with the positive axis,$\cos \gamma > 0$).Therefore,$\gamma = \frac{\pi}{3}$.

If a line makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $x$ and $y$-axes respectively,then the angle made by the line with the positive $z$-axis is

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