![\"A](\"http:\/\/daylateanddollarshort.com\/bloog\/wp-content\/uploads\/2012\/07\/wxyz-rct.png\" "\\"wxyz-rct\\"")
<\/a><\/div>\n$$\\cos\\frac{W}{2} = \\cos\\frac{X}{2} \\cos\\frac{Y}{2} \\cos\\frac{Z}{2} \\pm \\sin\\frac{X}{2} \\sin\\frac{Y}{2} \\sin\\frac{Z}{2}$$<\/p>\n
where, throughout, “\\(\\pm\\)” is “\\(+\\)” in hyperbolic space, and “\\(–\\)” in spherical space. Naturally, this leads to a Law of Cosines:<\/p>\n
$$\\begin{align}\\cos\\frac{W}{2} = \\cos\\frac{X}{2} \\cos\\frac{Y}{2} \\cos\\frac{Z}{2} &\\pm \\sin\\frac{X}{2} \\sin\\frac{Y}{2} \\sin\\frac{Z}{2}S \\\\[0.5em] &+ \\cos\\frac{X}{2}\\sin\\frac{Y}{2}\\sin\\frac{Z}{2} \\cos DA \\\\[0.5em] &+ \\sin\\frac{X}{2} \\cos\\frac{Y}{2} \\cos\\frac{Z}{2} \\cos DB \\\\[0.5em] &+ \\sin\\frac{X}{2} \\sin\\frac{Y}{2} \\cos\\frac{Z}{2} \\cos DC \\end{align}$$<\/p>\n
where<\/p>\n
$$S^2 := 1 – 2 \\cos DA \\cos DB \\cos DC – \\cos^2 DA – \\cos^2 DB – \\cos^2 DC$$<\/p>\n
And, as in the Euclidean case, this Law —which I call “First”— gives rise to a version —“Second”— that involves opposing dihedral angles and invites introduction of “pseudoface” elements.<\/p>\n
$$\\cos\\frac{W}{2} \\cos\\frac{X}{2} \\pm \\sin\\frac{W}{2} \\sin\\frac{X}{2} \\cos BC = \\cos \\frac{H}{2} = \\cos\\frac{Y}{2} \\cos\\frac{Z}{2} \\pm \\sin\\frac{Y}{2} \\sin\\frac{Z}{2} \\cos DA$$<\/p>\n
(Actually, I began calling the opposing dihedral version\u00a0without<\/em> the pseudoface element the “Second Law”, and the version\u00a0with<\/em> the pseudoface element the “Second-and-a-Halfth Law”; this phrasing persists in a couple of my notes. Nowadays, I just say “Second Law” and include the pseudo faces.)<\/p>\n