Each of the angles beta and gamma that a give | vedclass

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(A) Let $\alpha, \beta, \gamma$ be the angles made by the line with the positive $x, y, z$-axes respectively. Given $\beta = \frac{\alpha}{2}$ and $\g

Vedclass · Accepted Answer

$\frac{3 \pi}{4}$

Vedclass · Answer

(A) Let $\alpha, \beta, \gamma$ be the angles made by the line with the positive $x, y, z$-axes respectively. Given $\beta = \frac{\alpha}{2}$ and $\gamma = \frac{\alpha}{2}$. The direction cosines of the line satisfy the relation $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Substituting the given values: $\cos^2 \alpha + \cos^2(\frac{\alpha}{2}) + \cos^2(\frac{\alpha}{2}) = 1$. $\cos^2 \alpha + 2 \cos^2(\frac{\alpha}{2}) = 1$. Using the identity $\cos^2(\frac{\alpha}{2}) = \frac{1 + \cos \alpha}{2}$,we get $\cos^2 \alpha + 2(\frac{1 + \cos \alpha}{2}) = 1$. $\cos^2 \alpha + 1 + \cos \alpha = 1$. $\cos^2 \alpha + \cos \alpha = 0$. $\cos \alpha(\cos \alpha + 1) = 0$. This implies $\cos \alpha = 0$ or $\cos \alpha = -1$. If $\cos \alpha = 0$,then $\alpha = \frac{\pi}{2}$,so $\beta = \frac{\pi}{4}$. If $\cos \alpha = -1$,then $\alpha = \pi$,so $\beta = \frac{\pi}{2}$. The possible values for $\beta$ are $\frac{\pi}{4}$ and $\frac{\pi}{2}$. The sum of all possible values of $\beta$ is $\frac{\pi}{4} + \frac{\pi}{2} = \frac{3 \pi}{4}$.

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y-$ and $z-$axes,respectively,is half of the angle that this line makes with the positive $x-$axis. Then the sum of all possible values of the angle $\beta$ is

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