Alan

The x-values differ by 5 and the y-values differ by 6, so using Pythagoras's theorem the distance between M and N is √(5² + 6²)

$${\sqrt{{{\mathtt{5}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{6}}}^{{\mathtt{2}}}}} = {\mathtt{7.810\: \!249\: \!675\: \!906\: \!654\: \!4}}$$

cos(A+B) = cos(A)cos(B) - sin(A)sin(B) so 

cos(3pi/5)cos(3pi/20) - sin(3pi/5)sin(2pi/20) = cos(3pi/5 + 3pi/20) or cos(15pi/20) or cos(3pi/4) or cos(135°)

cos(135°) = -√2/2

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{135}}^\circ\right)} = -{\mathtt{0.707\: \!106\: \!781\: \!187}}$$

If I've understood the question correctly (that's a big if!) then there are results listed below for some small values of x.  I constructed a numerical algorithm to calculate these values.  I haven't been able to think of a generalisation (except when i = 1, in which case y_1av = (x-1)/2).

I suspect x is meant to be the number of items purchased here. That is:

cost of apples = 3*number of apples + 6 pesos.

cost of oranges = 4*number of oranges - 5 pesos

If that is the case then total cost = 3*12 + 6 + 4*6 - 5 = 61 pesos

405° is 45° beyond 360° and tan(45°) = 1, so tan(405°) = 1.

13pi/3 is pi/3 beyond 4pi (or 2 complete rotations; i.e. 2 rotations of 360°).  pi/3 is the same as 60° and cot(60°) = 1/√3, so cot(13pi/3) = 1/√3  (=√3/3).

Here you have a right-angled triangle with the opposite side = 90ft and the adjacent side = 140ft, so you need to use the inverse tangent function  (as tan(A) = opposite/adjacent)

$${\mathtt{A}} = \underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{90}}}{{\mathtt{140}}}}\right)} \Rightarrow {\mathtt{A}} = {\mathtt{32.735\: \!226\: \!272\: \!108^{\circ}}}$$

Angle of elevation ≈ 32.7°

Well, if you really mean this, then each component would be 100*cos(30°) newtons.  However, you would normally resolve a force into two directions that were perpendicular to each other, in which case one component would be 100*cos(30°) newtons and the other would be 100*sin(30°) newtons.

Here's an approach using the constant acceleration kinetics equations: