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 -axes. Then the sum of all possible values of the angle $\beta$ is

Question

Aicharya · Accepted Answer

$\begin{aligned}
& \beta=\frac{\alpha}{2}, \gamma=\frac{\alpha}{2} \\ & \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \cos ^2 \alpha+2 \cos ^2 \frac{\alpha}{2}=1 \\ & \cos ^2 \alpha+\cos \alpha=0 \\ & \cos \alpha(\cos \alpha+1)=0 \\ & \cos \alpha=0,-1 \\ & \alpha=\frac{\pi}{2}, \pi
\end{aligned}$
Now $\beta=\frac{\alpha}{2} \Rightarrow \frac{\pi}{4}, \frac{\pi}{2}$
so sum is $\frac{3 \pi}{4}$ *

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive y - and z -axes,…

Solution