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

<p>Given:</p> <p>Each of the angles $\beta$ and $\gamma$ is half of the angle that the line makes with the positive $x$-axis, i.e., $\beta = \gamma = \frac{\alpha}{2}$.</p> <p>The equation for the direction cosines of angles a line makes with the coordinate axes is given by:</p> <p>$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $</p> <p>Since $\beta = \gamma$, we substitute to get:</p> <p>$ \cos^2 \alpha + 2 \cos^2 \beta = 1 $</p> <p>Substitute $\cos \beta = \cos \gamma$:</p> <p>$ \cos \alpha = 2 \cos^2 \beta - 1 $</p> <p>Replacing back into the equation:</p> <p>$ (2 \cos^2 \beta - 1)^2 + 2 \cos^2 \beta = 1 $</p> <p>Simplify:</p> <p>$ (2 \cos^2 \beta - 1)(2 \cos^2 \beta + 1) = 0 $</p> <p>Solving for $\cos^2 \beta$, we have:</p> <p>$ 2 \cos^2 \beta - 1 = 0 \quad \text{or} \quad 2 \cos^2 \beta + 1 = 0 $</p> <p>The latter gives no real solutions, thus:</p> <p>$ \cos^2 \beta = \frac{1}{2} $</p> <p>Therefore, $\beta = \frac{\pi}{4}$ or $\beta = \frac{\pi}{2}$.</p> <p>Thus, the sum of all possible values of $\beta$ is:</p> <p>$ \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4} $</p>