A ray makes angles frac{pi}{3} and frac{pi}{4 | vedclass

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(A) Let $\alpha, \beta, \gamma$ be the angles made by the ray with the $X, Y$,and $Z$-axes respectively. The direction cosines of the ray are $\cos \a

Vedclass · Accepted Answer

$\frac{\sqrt{3}}{2}$

Vedclass · Answer

(A) Let $\alpha, \beta, \gamma$ be the angles made by the ray with the $X, Y$,and $Z$-axes respectively. The direction cosines of the ray are $\cos \alpha, \cos \beta, \cos \gamma$. We know that $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Given $\beta = \frac{\pi}{3}$ and $\gamma = \frac{\pi}{4}$. Substituting these values: $\cos^2 \alpha + \cos^2(\frac{\pi}{3}) + \cos^2(\frac{\pi}{4}) = 1$ $\cos^2 \alpha + (\frac{1}{2})^2 + (\frac{1}{\sqrt{2}})^2 = 1$ $\cos^2 \alpha + \frac{1}{4} + \frac{1}{2} = 1$ $\cos^2 \alpha + \frac{3}{4} = 1$ $\cos^2 \alpha = 1 - \frac{3}{4} = \frac{1}{4}$ Since $\sin^2 \alpha = 1 - \cos^2 \alpha$,we have: $\sin^2 \alpha = 1 - \frac{1}{4} = \frac{3}{4}$ Therefore,$\sin \alpha = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$.

$A$ ray makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$ and $Z$-axes respectively. Then,the value of the sine of the angle made by the ray with $X$-axis is

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