Goal rearranging, cot(θ2)=x for θ.
Since most calculators or other aids don't have a "cot" button or a cot−1 or arccot OR acot button1 (different word for the inverse cotangent function, cot backward), we're going to do this in terms of tan.
cot(θ2)=1tan(θ2) leaving us with
1tan(θ2)=x .
Now we take one over both sides.
11tan(θ2)=1x , which goes to
tan(θ2)=1x .
At this point we need to get the θ 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(45o)=1. arctan takes a ratio and produces an angle arctan(1)=45o 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(θ2))=arctan(1x) which becomes
θ2=arctan(1x) and finishing up we get
θ=2⋅arctan(1x).
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−1x can seem like 1tanx, while atanx and arctanx 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 −180o and 180o, to use other angles you need to use your brain!
