A second quadrant angle is between π/2 and π (90° and 180°).
sinA=2sin(A/2)cos(A/2)=¼.
4sin²(A/2)cos²(A/2)=4sin²(A/2)(1-sin²(A/2))=1/16.
64sin²(A/2)-64sin⁴(A/2)=1, 64sin⁴(A/2)-64sin²(A/2)+1=0
Let y=sin²(A/2), 64y²-64y+1=0.
We can solve this using the formula: y=(64±√(4096-256)/128=(4±√15)/8.
So sin(A/2)=√((4±√15/8). This gives us two different angles.
We know that sinA=0.25, so A=2.8889 radians which is between π/2=1.57 radians, and π=3.14 radians.
Therefore a rough estimate of A/2 is 1.4445 radians, and sin(1.4445)=0.9920.
When we find the two sine values from above we get 0.9920 and 0.1260.
From this we accept only sin(A/2)=√((4+√15)/8) as the exact value where A is in the second quadrant.