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 $\alpha, \beta, \gamma$

Vedclass · Accepted Answer

$\frac{\pi}{3}$

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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 relation:$\cos^2 \left(\frac{\pi}{3}\right) + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1$$\left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \gamma = 1$$\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 = \pm \frac{1}{2}$Since $\gamma$ is the angle between $0$ and $\pi$,$\cos \gamma = \frac{1}{2}$ implies $\gamma = \frac{\pi}{3}$ (or $\cos \gamma = -\frac{1}{2}$ implies $\gamma = \frac{2\pi}{3}$).Comparing with the options,the correct angle is $\frac{\pi}{3}$.

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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