Okay, so I did this; a basic example of 4:3 csgo hipfire: 2.4 @ 400dpi, 1.2 @ 800, & 0.6 @ 1600, et cetera:
a=3/4; x=90; θ=2 arctan(a tan((π x)/360)); (C csc(θ/2))/sqrt(2)=43.2955 cm;
http://www.wolframalpha.com/input/?i=a%3D3%2F4;+x%3D90;+θ%3D2+arctan(a+tan((π+x)%2F360));+(C+csc(θ%2F2))%2Fsqrt(2)%3D43.2955;
And C solved to 36.7374 cm.
I then try to match hipfire viewspeed to the first awp zoom with my C I found:
a=3/4; x=40; θ=2 arctan(a tan((π x)/360)); (36.7374 csc(θ/2))/sqrt(2);
http://www.wolframalpha.com/input/?i=a%3D3%2F4;+x%3D40;+θ%3D2+arctan(a+tan((π+x)%2F360));+(36.7374+csc(θ%2F2))%2Fsqrt(2);
Which got me 98.6445 cm. However the alpha calculator shows 98.6446 cm, which probably means that the calculator just rounds up at the ten thousandths place.
Just to check, I put my C back into the formula that handles hipfire, a=3/4; x=90; θ=2 arctan(a tan((π x)/360)); (36.7374 csc(θ/2))/sqrt(2);
http://www.wolframalpha.com/input/?i=a%3D3%2F4;+x%3D90;+θ%3D2+arctan(a+tan((π+x)%2F360));+(36.7374+ csc(θ%2F2))%2Fsqrt(2)
I get 43.2954 cm, but it should be 43.2955. Just more minor discrepancies I suppose.
For the sake of redundancy I'll try your other way you recommended with the random dpi:
a=3/4; x=90; h=1080; m=130.611; C=(sqrt(2) π h)/m; θ=2 arctan(a tan((π x)/360)); (C csc(θ/2))/sqrt(2);
http://www.wolframalpha.com/input/?i=a%3D3%2F4;+x%3D90;+h%3D1080;+m%3D130.611;+C%3D(sqrt(2)+π+h)%2Fm;+θ%3D2+arctan(a+tan((π+x)%2F360));+(C+csc(θ%2F2))%2Fsqrt(2)
The above got me approximately 43.2955 cm.
Then I isolated h, m, c to see if I get the same C from earlier, 36.7374 cm.
http://www.wolframalpha.com/input/?i=h%3D1080;+m%3D130.611;+C%3D(sqrt(2)+π+h)%2Fm;
I got approximately 36.7374 as my C.
From this exercise I have a few questions:
1. Does variable 'a' have to equal '3/4' when said game uses strict horizontal 4:3 fov handling?
2. Otherwise does 'a' have to 'match' the screen's aspect ratio when using the other formulas?
3. Can I use the same 'C' in other games, or will I have to check for 'C' for every game?