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!
