Wednesday, September 15, 2021

Sum of angles: Ptolemy

 The "sum of angles" theorems are incredibly useful in calculus.  These are formulas for the sine (or cosine) of the sum (or difference) of two angles.  There are a number of derivations, of which my favorite is a proof-without-words (here).

Recently, I got interested in Ptolemy's theorem, which is illustrated in this graphic:

There are proofs based on similar triangles (a bit complicated), and one based on areas with a wonderful trick in the middle.  Another terrific proof-without-words for that is here.

It turns out that there are easy proofs for the trigonometric theorems starting from Ptolemy's theorem, and I want to look at them here.  The first one is for sin s+t:

The proof depends on the following details.  First, the black line in the middle is a diameter of the circle, scaled to have length 1.  As a result, the two triangles are right triangles, justifying the labels on the sides.  Ptolemy says to multiply two pairs of opposite sides and add them.

In addition, the dotted line is the chord of the sum of the two angles, and its length is:  2R sin s+t = L.  But 2R is scaled to be 1, so finally we have the formula for the sine of s+t.  It basically writes itself.

The second one is for the difference.  This is extra, since the easiest way to get from the sum to the difference is to use the fact that cosine is an even function and sine is odd:  sin(x) = -sin(-x).

If you're trying to set this up yourself, a big clue is that since the formula has a minus sign in it (sin s cos t minus sin t cos s), the sin s-t term is going to be one of the sides, and naturally, it is the side opposite the diameter.

As for the cosine formulas, there is yet another trick.  We will still have that the dotted line is going to be the sine of some angle θ, but it is also the cosine of the angle that is complementary to θ!  And again, cosine s+t is one of the sides.

The last one isn't quite as pretty (I don't know if there are others, I came up with this one myself).  The hard part is writing an expression for the angle marked with the black dot, but its complement is just s-t.

I still recommend the following as the best way to re-derive the formulas if you're stuck:

And finally, this is the idea of how to carry out the proof-without-words for Ptolemy's theorem.  Cut the original figure alone one diagonal, lay the two triangles side-by-side, rescale, check all the angles, and then find the corresponding triangle to complete the parallelogram, from the original figure.

For more details, you can check out my books on geometry on github.