If a line makes angles frac{pi}{4} and frac{p | vedclass

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(B) Let the line make angles $\alpha, \beta, \gamma$ with the $X, Y, Z$-axes respectively. The direction cosines are $l = \cos \alpha, m = \cos \beta,

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$\frac{2 \pi}{3}$

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(B) Let the line make angles $\alpha, \beta, \gamma$ with the $X, Y, Z$-axes respectively. The direction cosines are $l = \cos \alpha, m = \cos \beta, n = \cos \gamma$. Given $\beta = \frac{\pi}{4}$ and $\gamma = \frac{\pi}{3}$. We know that $l^2 + m^2 + n^2 = 1$. Substituting the values: $\cos^2 \alpha + \cos^2 \frac{\pi}{4} + \cos^2 \frac{\pi}{3} = 1$. $\cos^2 \alpha + (\frac{1}{\sqrt{2}})^2 + (\frac{1}{2})^2 = 1$. $\cos^2 \alpha + \frac{1}{2} + \frac{1}{4} = 1$. $\cos^2 \alpha + \frac{3}{4} = 1 \Rightarrow \cos^2 \alpha = \frac{1}{4}$. $\cos \alpha = \pm \frac{1}{2}$. Since the angle $\alpha$ is obtuse,$\cos \alpha$ must be negative,so $\cos \alpha = -\frac{1}{2}$. Therefore,$\alpha = \pi - \frac{\pi}{3} = \frac{2 \pi}{3}$.

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

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