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

Accepted Answer

Answer: C. \frac{3 \pi}{4} Given: 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}. The equation for the direction cosines of angles a line makes with the coordinate axes is given by: \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 Since \beta = \gamma, we substitute to get: \cos^2 \alpha + 2 \cos^2 \beta = 1 Substitute \cos \beta = \cos \gamma: \cos \alpha = 2 \cos^2 \beta - 1 Replacing back into the equation: (2 \cos^2 \beta - 1)^2 + 2 \cos^2 \beta = 1 Simplify: (2 \cos^2 \beta - 1)(2 \cos^2 \beta + 1) = 0 Solving for \cos^2 \beta, we have: 2 \cos^2 \beta - 1 = 0 \quad 	ext{or} \quad 2 \cos^2 \beta + 1 = 0 The latter gives no real solutions, thus: \cos^2 \beta = \frac{1}{2} Therefore, \beta = \frac{\pi}{4} or \beta = \frac{\pi}{2}. Thus, the sum of all possible values of \beta is: \frac{\pi}{4} + \frac{\pi}{2} = \frac{3\pi}{4}

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

Answer

Explanation

More JEE Math questions