Question 959220
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Find exact Value of tan(Beta/2), Given that tan(Beta) = square root 5/2 and pi < Beta < 3pi/2
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculations in the post by @lwsshak3 are fatally and totally incorrect.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is enough to notice that Beta is in QIII (given), hence, Beta/2 is in QII,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;so tan(Beta) must be negative, while @lwsshak3 gives a positive number as the answer.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;His checking procedure also is wrong.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below is my correct solution.



use x for Beta
tan(x) = √5/2 (given, so BETA is in QIII)
hypotenuse of reference right triangle in quadrant III = √(√5)^2+2^2)=√(5+4)=3
sin(x) = -√5/3 in QIII (negative)
cos(x) = -2/3 in QIII (negative)
tan(x/2) = sin(x)/(1+cos(x))
tan(x/2) = (-√5/3)/(1-2/3) = (-√5/3)/(1/3) = -√5/1 = -√5.


Check:


tan(x) = √5/2 in QIII
x = 48.19° + 180° = 228.19°
x/2 ≈ 114.095°
tan(x/2) ≈ tan(114.095°) ≈ -2.236
exact value = -√5 ≈ -2.236


Solved correctly and checked in a right way, too.