Each of the angles and that a given line makes with the positive y-axes 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 is [2025]

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Solution

Q.
Each of the angles $β$ and $γ$ that a given line makes with the positive y-axes 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 [2025]

1 $π$

2 $\frac{π}{2}$

3 $\frac{3 π}{4}$

4 $\frac{3 π}{2}$

Ans.
(3)

Let the line makes angle $α$ with positive x-axis, then $β = \frac{α}{2}$ and $γ = \frac{α}{2}$

Now, $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$

$\Rightarrow \cos^{2} α + 2 \cos^{2} \frac{α}{2} = 1$

$\Rightarrow \cos^{2} α + \cos α = 0$

$\Rightarrow \cos α (\cos α + 1) = 0$

$\Rightarrow \cos α = 0, - 1 \Rightarrow α = \frac{π}{2}, π$

Now, $β = \frac{α}{2} \Rightarrow β = \frac{π}{4}, \frac{π}{2}$

So, required sum $= \frac{π}{4} + \frac{π}{2} = \frac{3 π}{4}$ .

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