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Archived Post

This post is archived from my Posterous blog, which shut down in 2012. Some posts have been edited slightly to fix typographical errors, correctly represent the gender of some individuals, and remove unnecessarily-gendered language. You can view the full archive here.

A Little Math Help?

Archive Post
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Anybody want to help me with some math? Consider:

A geometric diagram showing two parallell lines and two intersecting lines at different angles. The point at which the intersecting lines meat is labeled with a star. The angles of the interseting lines against the parallel lines are labels "A" and "A-question-mark".

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:

A mathematical formula.

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:

Another diagram similar to the one above, but the angles are negative.

A working case with negative angles.

But this doesn’t work…

Another diagram similar to the one above, but the angles are now larger than 90 degrees.

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?