My problem lies with the known indicated airspeed, which is 125 KIAS. Can anyone point me to how 125 KIAS becomes 127 KIAS?
It's worth noting that your climb gradient is approximately 20% or 12 degrees. This is not really a "small angle" for trigonometric purposes. It would be ok for in-flight approximations, but not for a written exam.
2590fpm x 60 / 6076 = 25.576 kts vertical speed.
Inverse Tan (0.2014) = 11.387 degrees.
25.576 / sin(11.387) = 129.5 kTAS.
Possible errors, in decreasing order of significance:
Pressure error (also known as position error)
Trigonometric error by dropping the sine
Rounding error in calculation
For in-flight purposes, it might look something like this:
My airspeed (IAS or TAS will do at sea level) is 125kts.
My required climb gradient is
just over 20%.
125 x 20 = 2500fpm.
I need to achieve
just over 2500fpm.