Let A, B, and C be the angles of a triangle. Given that tan (A/2) = 1/4 and tan (B/2) = , find tan (C/2).
Using the half-angle formulae:
.........(1)
..........(2)
........(*)
Since the angles in a triangle sum to 180 degrees, then:
(Take tan of both sides).
Thus, consider: ........(**)
Now consider , and using (1) and (2) gives:
......(***)
Thus, substituting (***) in (**) gives:
Next, consider (*) and let
Then,
Then,
Thus,
But, C is a positive angle (Angle in a triangle), hence
Therefore, only is the desired solution.
Thus,
Hope this helps, and I think there is a simpler solution than this!
Using the half-angle formulae:
.........(1)
..........(2)
........(*)
Since the angles in a triangle sum to 180 degrees, then:
(Take tan of both sides).
Thus, consider: ........(**)
Now consider , and using (1) and (2) gives:
......(***)
Thus, substituting (***) in (**) gives:
Next, consider (*) and let
Then,
Then,
Thus,
But, C is a positive angle (Angle in a triangle), hence
Therefore, only is the desired solution.
Thus,
Hope this helps, and I think there is a simpler solution than this!