Beginner's Guide to Propulsion: Forces in a Climb - Vectors

Using the first equation: L cos (c) + F sin (c) -D sin (c) - W = ma (vertical)
and the third equation: F -D = F_ex , explain and show how to obtain
the fourth equation: L cos (c) + F_ex sin (c) - W = ma (vertical).

Factor sin (c) out of last two terms and substitute F_ex for (F - D).
Using the second equation and the third equation, explain and show how to obtain the fourth equation.
In the second equation, factor cos (c) out of the first and third terms and substitute as in above.
Why isn't W multiplied by a sine or cosine?
Weight is a downward force, so it would be W sin (90) or just W. (sin 90 is 1).
Using the given diagram, theorems/postulates of geometry, and vectors, explain each term in the first two equations.

Vertical Component.
1. V₁ = F sin 10 positive (upward) vertical
  V₂ = L cos 10 positive vertical
  1. angle 1 + angle 2 = 90 Lift is perpendicular (vertical) to flight path
    angle 1 + angle 3 = 90 Acute angles of a right triangle are complementary.
    
    Therefore angle 1 + angle 2 = angle 1 + angle 3
    
    Then angle 2 = angle 3
    
    Since angle 2 = 10 then angle 3 must also.
  V₃= D sin 10 negative (downward) vertical
  
  V₄ = W
Horizontal Component
1. H₁= F cos 10 Positive (forward)
  H₂ = L sin 10 Negative (backward)
  
  H₃= D cos 10 Negative (backward)
  
  "W" only affects the vertical, not horizontal, or W cos 90 = 0