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 -axis. Then the sum of all possible values of the angle β is:
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Step-by-Step Solution
Step 1: Define direction cosines and angles
Let the angles that the line makes with the positive x -, y -, and z -axes be α, β, and γ respectively. The direction cosines of the line are cosα, cosβ, and cosγ. A fundamental property of direction cosines is that the sum of their squares is equal to 1.
Step 2: Express angles in terms of β
According to the problem statement, the angle β (with the y -axis) and γ (with the z -axis) are each half of the angle α (with the x -axis). This means β=2α and γ=2α. We can rewrite these relationships as α=2β and γ=β.
Step 3: Substitute into the direction cosine identity
Now, substitute the expressions for α and γ in terms of β into the direction cosine identity. This gives us an equation solely in terms of β.
Step 4: Simplify and solve for cosβ
Using the double angle identity cos(2β)=2cos2β−1, we can simplify the equation. The equation becomes (2cos2β−1)2+2cos2β=1. Expanding and rearranging terms, we get 4cos4β−4cos2β+1+2cos2β=1, which simplifies to 4cos4β−2cos2β=0. Factoring out 2cos2β yields 2cos2β(2cos2β−1)=0.
Step 5: Find possible values of cosβ
From the simplified equation 2cos2β(2cos2β−1)=0, we have two possibilities: either 2cos2β=0 or 2cos2β−1=0. If 2cos2β=0, then cosβ=0. If 2cos2β−1=0, then 2cos2β=1, so cos2β=21, which means cosβ=±21.
Step 6: Determine possible values of β
For cosβ=0, the possible value for β in the range [0,π] is 2π. For cosβ=21, β=4π. For cosβ=−21, β=43π. These are the possible values for β since angles with axes are typically considered in the range [0,π]. We must also check that α=2β is also in this range. For β=2π, α=π. For β=4π, α=2π. For β=43π, α=23π, which is outside the range [0,π]. Therefore, β=43π is not a valid solution.
Step 7: Calculate the sum of possible values of β
The valid possible values for β are 2π and 4π. Summing these values gives the final answer.