[VA] SRC#004- Fun with Sexagesimal Trigs

[J-F Garnier]

(02-12-2019 12:14 PM)Albert Chan Wrote:  Prove A = 2^29 = 536870912:

Checking the edges, using shorthand t# = tan(#°):

tan(A°+B°) = (tA + tB) / (1 - tA tB)

(t60 + tA) (t60 + tB), where A+B=30°
= t60^2 + t60*(tA + tB) + tA tB
= t60^2 + t60*t30*(1 - tA tB) + tA tB
= 3 + 1 - tA tB + tA tB
= 2^2

...

14 pairs of edges and 1 center, all can considered value of 2, thus A = 2^29

Great!

With the same method (and notations), it's easy to prove D = 2^22:

tan(A°+B°) = (tA + tB) / (1 - tA tB)

(1 - 1/tA) (1 - 1/tB), where A+B=45°
= 1 - (1/tA + 1/tB) + 1/(tA tB)
= 1 - (tA + tB) / (tA tB) + 1 / (tA tB)
= 1 - t45 (1 - tA tB) / (tA tB) + 1 / (tA tB)
= 1 - (1 / (tA tB) - 1) + 1 / (tA tB)
= 2

J-F