The direction cosines of the line making angl | vedclass

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Question

(A) Let the angles made by the line with the $X, Y$ and $Z$ axes be $\alpha = \frac{\pi}{4}$,$\beta = \frac{\pi}{3}$,and $\gamma = 	heta$. The direct

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$

Vedclass · Answer

(A) Let the angles made by the line with the $X, Y$ and $Z$ axes be $\alpha = \frac{\pi}{4}$,$\beta = \frac{\pi}{3}$,and $\gamma = 	heta$. The direction cosines are given by $l = \cos \alpha$,$m = \cos \beta$,and $n = \cos \gamma$. We know that $l^2 + m^2 + n^2 = 1$. Substituting the values: $(\cos \frac{\pi}{4})^2 + (\cos \frac{\pi}{3})^2 + \cos^2 	heta = 1$. $(\frac{1}{\sqrt{2}})^2 + (\frac{1}{2})^2 + \cos^2 	heta = 1$. $\frac{1}{2} + \frac{1}{4} + \cos^2 	heta = 1$. $\frac{3}{4} + \cos^2 	heta = 1$. $\cos^2 	heta = 1 - \frac{3}{4} = \frac{1}{4}$. Since $0 Thus,the direction cosines are $\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$,$\cos \frac{\pi}{3} = \frac{1}{2}$,and $\cos 	heta = \frac{1}{2}$. Therefore,the direction cosines are $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$.

The direction cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta$ $(0 < \theta < \frac{\pi}{2})$ respectively with $X, Y$ and $Z$ axes are:

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