Each of the angles 
and 
that a given line makes with the positive 
- and 
-axes, respectively, is half of the angle that this line makes with the positive 
-axes. Then the sum of all possible values of the angle 
is
Text Solution
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Given:
Each of the angles 
and 
is half of the angle that the line makes with the positive 
-axis, i.e., 
.
The equation for the direction cosines of angles a line makes with the coordinate axes is given by:
Since 
, we substitute to get:
Substitute 
:
Replacing back into the equation:
Simplify:
Solving for 
, we have:
The latter gives no real solutions, thus:
Therefore, 
or 
.
Thus, the sum of all possible values of 
is:
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