SOLUTION: If tan x > 0, then tan (x/2) > 0. True or False?

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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): You can put this solution on YOUR website!
If tan x > 0, then tan (x/2) > 0. True or False?
-------------
For example, tan(200) > 0
tan(100) < 0
False
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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