Each of the angles and that a given line makes with the positive y- and z-axes, respective…

Question

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 $\alpha$ that this line makes with the positive $x$-axis. Then the sum of all possible values of the angle $\beta$ (equivalently, all valid solution branches of the governing equation below) is:

Accepted Answer

With $\alpha=2\beta=2\gamma$ and $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$:
$$\cos^2\alpha+2\cos^2\beta=1.$$
Using $\cos\alpha=2\cos^2\beta-1$:
$$(2\cos^2\beta-1)^2+2\cos^2\beta=1 \implies (2\cos^2\beta-1)(2\cos^2\beta+1)=0.$$
Since $2\cos^2\beta+1\neq0$, we need $2\cos^2\beta-1=0\Rightarrow\cos^2\beta=\dfrac12$, giving the valid solution branches $\beta=\dfrac{\pi}{4}$ and $\beta=\dfrac{\pi}{2}$.

Sum $=\dfrac{\pi}{4}+\dfrac{\pi}{2}=\dfrac{3\pi}{4}$.

Solutions · 1