Triple Trig Threat!

we get that $\begin{cases} \alpha +\beta -\gamma=\sin^{-1} \dfrac{1}{2}=30\\ -\alpha +\beta +\gamma=\cos^{-1} \dfrac{1}{2}=60\\ \alpha -\beta +\gamma=\tan^{-1} 1=45 \end{cases}$ now add all 3 to get $\alpha +\beta +\gamma=135$

According to given condition that all angles are acute means sum of all three angles should be in between 0 and 267 degrees (0<=A+B+C<=267) if we assume that all angles are in integer form (Min value of angle is 0' or Max is 89') and sum of two angles minus other one angle is always in between -89 and 178 degrees. Here negative angle is not considered so that is in between 0 and 178 degrees.

for SINE

sin(A+B-C) = 1/2 means A+B-C can be 30 or 150 degree because sine is positive in 1st(0 to 90) and 2nd quadrant(90 to 180) and to satisfy the condition of acute angle.

similarly for COSINE

cos(B+C-A) = 1/2 => B+C-A = 60 degree because cosine is positive in 1st and 4th quadrant(270 to 360) so we take only 1st quadrant to satisfy the condition of acute angle.

similarly for TAN

tan(A+C-B) = 1 => A+C-B = 45 degree because tan is positive in 1st and 3rd quadrant(180 to 270) so we take only 1st quadrant to satisfy the condition of acute angle.

now,

           A+B-C = 30 (not 150)

           B+C-A = 60

           A+C-B = 45

Add the three equations and get the sum of three angles as 135 degree

Simply A+B-C = 30 , B+C-A=60 A+C-B=45 Add all three to get A+B+C = 135