First, we must determine all the forces acting on this plane other than the forces generated by its engine.
There are two such forces: gravity and resistance.

In the answer at the back, they show that when you add these two vectors tail-to-tip, you get a resultant vector that is 30 degrees with respect to the horizontal downwards. Therefore, the engine must exert a force in the opposite direction to keep the plane flying straight, i.e. 30 degrees upwards. Why won't the plane elevate? because if you break the vectors down into components, you will see that the forces in the vertical direction cancel out.

In terms of magnitude, they calculated this using the 30-60-90 triangle. To determine the magnitude, you apply the same length ration to the values you are calculating. Air resistance is 8.7 x 10^4, gravity is 5 x 10^4, resultant force is 10 x 10^4. If you prefer not to apply the ratios, you could have just done trig and found that

"F" = (5 x 10^4)/sin30 = (5 x 10^4)/(1/2) = 10 x 10^4