I have seen some answers to similar questions, but usually they are satisfied with just limiting the plot using PlotRange. I want to keep the plot range as I have it, and just limit the range that is evaluated. I don't want to show that horizontal line at 90 (which is part of the output) so I want to limit the output to just above 90 (like 90.1) so that I can instead show the horizontal dashed line as the output approaches it asymptotically. I think it does not tell the story as clearly if I just limit the range to 90 since it would be approaching the x-axis so I want to keep the displayed range down to 80.
ClearAll["Global`*"]
eq1 = NSolve[0.5==(-\[Pi]/3)r^3(1/Cos[\[Theta]*Degree]^3)(2+Sin[\[Theta]*Degree])(1-Sin[\[Theta]*Degree])^2, \[Theta]];
Labeled[Plot[\[Theta]/.eq1, {r,0.6203, 6} ,ImageSize->700,ImageMargins->0,ImagePadding->{{22, 10}, {15, 5}}, PlotRange-> {{0,6},{90,180}}, GridLines ->{{{0.62,Dashed}}, {{90, Dashed}}}, Ticks ->{{0,0.62,1,2,3,4,5,6},{10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180}}],{"Contact Angle (\[Degree])","Tube Radius", "Maximum Contact Angle for Plug of Volume 1 at Critical Bond Number" }, {Left, Bottom, Top}, RotateLabel -> True, Spacings ->{0,1,0}]