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

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(D) We know that the direction cosines of a line satisfy the relation: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$, where $\alpha, \beta, \gamm

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$\frac{\pi}{3}$

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(D) We know that the direction cosines of a line satisfy the relation: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$, where $\alpha, \beta, \gamma$ are the angles made by the line with the $X, Y,$ and $Z$-axes respectively. Given $\alpha = \frac{\pi}{3}$ and $\beta = \frac{\pi}{4}$. Substituting these values into the identity: $\cos^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1$ $\Rightarrow \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \gamma = 1$ $\Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1$ $\Rightarrow \frac{3}{4} + \cos^2 \gamma = 1$ $\Rightarrow \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4}$ $\Rightarrow \cos \gamma = \pm \frac{1}{2}$. Considering the principal value, $\cos \gamma = \frac{1}{2} = \cos \left(\frac{\pi}{3}\right)$, which gives $\gamma = \frac{\pi}{3}$.

If a line makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the $X$-axis and $Y$-axis respectively, then the angle made by the line with the $Z$-axis is

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