Why is the tangent of 22.5 degrees not 1/2? Sorry for the stupid question, but why is the tangent of 22.5 degrees not 1/2? 
(Okay... I get
 that that the tangent of 45 degrees is 1 ("opposite" =1, "adjacent" =1, 1/1 = 1.  Cool.  I am good with that.) Along those same lines, if the "opposite" drops to 1/2 relative to the "adjacent" i.e., "opposite" = 1, "adjacent" = 2 therefore 1/2. What am I missing?  Thanks in advance for your help.
It helps to realize that the tangent of an angle can be viewed as the length of a segment tangent to the unit circle (hence the name):



As the above helps illustrate, as an angle sweeps from 0 to 90 degrees, the corresponding value of tangent ranges from 0 to infinity. Behavior that exotic cannot reasonably be expected to follow a simple rule like "halving the angle halves the tangent"; just look at the reverse: Doubling a $45^\circ$ angle does a lot more than double its tangent value!

I'll leave you to ponder the deeper implications.
The basic error being made is the assumption that you are converting one unit into another unit such as when you convert meters into yards.  Degrees are units but tangent represents a ratio.