Mathematica gives an explicit formula for the solution to your equation, if I do not misunderstand your third equation:

eq1 = 1.30000 == 
   Sqrt[(6.64730 - x)^2 + (10.17180 - y)^2 + (5.66580 - z)^2];
eq2 = w == Sqrt[(5.9169 - x)^2 + (11.4014 - y)^2 + (4.4454 - z)^2];
eq3 = 118.0000 == 
   ArcCos[((1.8800)^2 + (1.30000)^2 - (w)^2)/(2*1.8800*(1.30000))]*(180/Pi);
sol = Solve[{eq1, eq2, eq3}] // Simplify 

The solution triples x,y,z lie in a 3D circle:

ParametricPlot3D[Evaluate[{x, y, z} /. sol], {x, 0, 20}, 
 AxesLabel -> {x, y, z}]

Your second equation is strange, but it looks like another sphere. Perhaps you are looking for points in the intersection of two spheres:

sphere1 = Sphere[{6.64730, 10.17180, 5.66580}, 1.3];
sphere2 = 
  Sphere[{5.9169, 11.4014, 4.4454}, 
   Sqrt[-(Cos[118 Pi/180] - (1.88^2 + 1.3^2))]];
Graphics3D[{sphere1, sphere2}]
RandomPoint[DiscretizeRegion[RegionIntersection[sphere1, sphere2]]]

HI Gianluca,

Well the variables are orthogonal coordinates which get converted to crystallographic coordinates and plugged into a program called teXsan which tells us the rotational movement of a dummy oxygen atom from 0-360 degrees on a molecule. Its really convoluted, but do you know how I would incorporate the second equation into the code that Nasser wrote?

I really just need something that will take the guesswork out of finding different combinations of those 3 orthogonal coordinates that produce a distance of 1.3 angstroms and 118 degrees using the aforementioned equations.

Hi Gianluca,

Can you explain what you did? I just started using mathematica an hour ago so I'm sorry if I sound vapid.

Also thanks for replying!

Hi Nasser,

Yes that was exactly what I was looking for thank you!! There is another layer to this problem though. So the three variables must =1.30000 but they must also equal 118.0000 using the following equation:

118.0000=ACOS[[[1.8800]^2 + [the value given by first equation]^2 - [=SQRT[[5.9169-x]^2+[11.4014-y]^2+[4.4454-z]^2]]^2] / [2 * 1.8800 * the value given by first equation ] ] * 180/3.1415926

If something doesn't make sense then my bad. He emailed me a copy of his excel spreadsheet and I'm trying to decipher how he set the formulas up. If you want to take a look at the excel spreadsheet I can PM it to you and tell you exactly what we are trying to do with this number crunching.