A ray makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$ and $Z$-axes respectively. Then, the value…
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
$\frac{\sqrt{3}}{2}$
$\frac{1}{2}$
$\frac{1}{\sqrt{2}}$
$1$
Solution
If $\alpha, \beta, \gamma$ are angles made by a ray with $X, Y$ and $Z$-axes respectively, then $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1$
$
\begin{aligned}
& \therefore \cos ^2 \alpha+\cos ^2\left(\frac{\pi}{3}\right)+\cos ^2\left(\frac{\pi}{4}\right)=1 \\
& \Rightarrow \cos ^2 \alpha=1-\frac{1}{4}-\frac{1}{2}=\frac{1}{4} \\
& \Rightarrow \cos \alpha= \pm \frac{1}{2} \\
& \therefore \sin \alpha=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}
\end{aligned}
$