Anybody want to help me with some math? Consider:

Diagram 1
I know angle A
, distance D
, angle AΔ
and distance DΔ
. I’m trying to find angle A
?. This is what I came up with:

The proposed formula.
At first I thought I was all good, but this doesn’t work for all A
and AΔ
(which is obvious in hindsight). For instance, this works fine if A
is negative:

A working case with negative angles.
But this doesn’t work…

A failing case with large angles.
…essentially because the range of arc tangent is (-90,+90)
degrees. Or, put another way, because there is no right triangle with angle A
.
Is there a single formula that will work for all A
and AΔ
without the need for conditionals?