Each of the angles β and γ 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 β is

Solution

Maths sir · 4 steps

Write everything in β, use the identity, then reject any root that makes α > π.

1. Set up. The line makes angle α with the x-axis, and β = γ = α/2. So α = 2β and γ = β. Put these into cos² α + cos² β + cos² γ = 1.

   cos² 2β + 2 cos² β = 1

2. Solve the quadratic. Let c = cos² β, so cos 2β = 2c - 1. Then (2c-1)² + 2c = 1 gives 4c² - 2c = 0, so c = 0 or c = 1/2. That means cos β = 0 or cos β = ±1/√2.
3. Check the range. Every direction angle lies in [0, π]. Since α = 2β must also lie in [0, π], β cannot exceed π/2.

   β	α = 2β	Valid?
   π/2	π	yes: direction cosines (-1, 0, 0)
   π/4	π/2	yes: direction cosines (0, 1/√2, 1/√2)
   3π/4	3π/2	no: α is more than π

4. So. The possible values are β = π/2 and β = π/4, and their sum is 3π/4. The answer is (c).

The idea to keep. Direction angles all live in [0, π]. After solving the identity, test each root against that range for every angle, not just the one you solved for.

Answer from the NTA final answer key. Solution by PrepWiser.