We cannot solve this without a right hand side, so I'm just going to go with x.
Goal rearranging, cot(\theta/2)=x for \theta.
Since most calculators or other aids don't have a "cot" button or a cot^{-1} or arc cot OR acot button""^1 (different word for the inverse cotangent function, cot backward), we're going to do this in terms of tan.
cot(\theta/2)=1/tan(\theta/2) leaving us with
1/tan(\theta/2)=x .
Now we take one over both sides.
1/{1/tan(\theta/2)}=1/x , which goes to
tan(\theta/2)=1/x .
At this point we need to get the \theta outside of the tan, we do this by taking the arctan, the inverse of tan. tan takes in an angle and produces a ratio, tan(45^o)=1. arctan takes a ratio and produces an angle arctan(1)=45^o ""^2. This means that arctan(tan(45))=45 and tan(arctan(1))=1 or in general:
arctan(tan(x))=x
and
tan(arctan(x))=x.
Applying this to our expression we have,
arctan(tan(\theta/2))=arctan(1/x) which becomes
\theta/2=arctan(1/x) and finishing up we get
\theta=2*arctan(1/x).
You my notice I used footnotes! there are some subtleties to inverse trig functions I chose to pack down here.
1) Names of inverse trig functions. The formal name of an inverse trig function is "arc"- trig function ie. arctan, arccos arcsin. This is shorted two ways, "atan", "acos" "asin" which is used in computer programming and math programs and the HORRIBLE "tan^-1", "sin^-1" "cos^-1" which is used in a lot of calculators. It is HORRIBLE because tan^-1 x can seem like 1/tan x, while atan x and arctan x is much much less likely to confuse a reader. Use atan or arctan in your algebra.
2) Since all values of tangent occur TWICE in the unit circle, arctan normally returns angle between -180^o and 180^o, to use other angles you need to use your brain!
