Question 1202696: If tan x > 0, then tan (x/2) > 0. True or False?
Found 4 solutions by Alan3354, math_tutor2020, greenestamps, ikleyn:
Answer by Alan3354(69443)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Adding to @Alan3354's answer, we can use a calculator to find that
tan(200) = 0.363970
tan(100) = -5.671282
Those decimal values are approximate. The calculator is in degree mode.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
If x is between 0 and pi/2, then x/2 is between 0 and pi/4. In that case, since tan(x) > 0 everywhere between 0 and pi/2, the statement is true.
But if x is between pi and 3pi/2, then tan(x) is again positive. But in this case, x/2 is between pi/2 and 3pi/4, so tan(x/2) < 0. So in this case tan(x) > 0 but tan(x/2) < 0, so the statement is false.
ANSWER: False
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! .
This problem is remarkable.
It is remarkable, because one single example (or "counter-example") is enough to disprove the statement.
And even if a student does not have a calculator, he (or she) must know
the table values of trigonometric functions.
In particular, tan(240°) =  (in 3rd quadrant),
while tan(240°/2) = tan(120°) =  (in 2nd quadrant).
This single fact disproves the statement.
ANSWER. False.
Solved.
Good problem to learn (may be, for the first time), how counter-examples work in Math.
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