Deriving the Formulas for Sightline and Resultant angles

Warning for adults: This post is primarily about math. If you're a math phobe, hate math, whatever, you should hit the eject button now! For an excellent, math-free approach to the same topic, go here. This warning is only for adults--teenagers can do this stuff easily.

I've been too busy making planes to dabble in chairmaking lately, but it's never far from my mind. Chairmakers and planemakers have a lot in common, and a number of folks--like Caleb James--do both at a very high level.
I first read about sightlines and resultant angles eight years ago, when I checked Drew Langsner's book out of the public library. In use, it's the simplest, most intuitive way to drill chair mortises. The only difficulty is coming up with the numbers. I know of five ways to do it:

• Langsner has tables in the back of his book, laying out the sightline and resultant angles from 0° to 45°.
• Peter Galbert shows a very intuitive approach to drawing and scaling the sightline. In the linked article, he omits finding the resultant angle, but I think you'll understand how to do it if you keep reading.
• Chris Schwarz is the creator of the math-free approach, using wire models, that I linked to above.
• Jameel Abrams shows how to find the angles in Sketchup.
• You can use the sector made by Acer-Ferrous Toolworks.

So, do you need a sixth approach? Probably not. But I do! I've always used the numbers in Langsner's book, but I hate relying on a table of calculations without understanding where they come from, and why they work. I need to know what's under the hood and how it was made--maybe you're the same. Plus, if the apocalypse comes and sweeps away all my books, I'll still be able to make Windsor chairs in the post-nuclear hellscape.

Edit: If you just want the formulas for sightline and resultant angles, skip to the bottom of the post, where I give the two formulas and walk through a brief example. If you want to understand how the formulas are derived, read on.
 
To derive the formulas, we only need a bit of high school (sophomore year) math. We need to know SOHCAHTOA, and the inverses of the three basic trig functions. If you've forgotten this stuff, brush up with the linked explanations. I'll wait.

Oh, you're back? Great. I'll start with Galbert's method of drawing the rake and splay. Imagine dropping a plumb line from the top of a chair leg down to the floor. You'll form two triangles, one in side view (rake) and one in front view (splay). Now imagine flopping those triangles down on the floor to get a two-dimensional view, and you'll get this:

 

Since the the height is the same in both views, I've labeled them both with an arbitrary unit of one. As you'll see, this will be key to finding our formulas.
Next, label the rake and splay angles, r for rake and s for splay. Draw the rectangle defined by the rake and splay axes, and add in a diagonal. This is the sightline. You can think of the lower right vertex as the point where the bottom of the chair leg would sit.

Now we're ready to find the lengths of the rake and splay lines in terms of r and s. Using the tangent identity,

and similarly,

Here's what we've got so far.

Now we need a way to find the sightline angle--let's call it l--and to find the length of the sightline in terms of our other measurements.
The sightline angle is easy. Using the tangent identity again,

File this away for later--we'll use it to find our actual sightline angles.
Finding the length of the sightline is a little more complicated. Since it's the hypotenuse of a triangle, we'll need our sine function:

Now do a little algebra to isolate the "sightline" variable, and you'll get

Here's what we've got so far.

The last thing we need is a way to find the resultant angle, which I'll label t°. Let's draw one more triangle, defined by the common height and the sightline.

Using the tangent identity,

And now we've got the two formulas we need. Let's get rid of the confusing letters and use words instead:

and

Now let's look at an example. To find the angles we need to build a chair, we just have to punch the numbers into a calculator, and use the inverse tan (tan ^-1 ) button. Let's say we want 12° rake and 14° splay. For the sightline,

Now plug .851 into the inverse tan function:

So 40.4° is the sightline angle.

For the resultant angle:

Plug .327 into the inverse tan function:

And 18.1° is the resultant angle. Now, go make some chairs!