| Spoiler:
Start with the identity that
sin 3y = 3 sin y - 4 (sin y)^3
If r1 = sin 10 degrees, r2 = sin 50 degrees, r3 = sin -70 degrees,
then
sin 30 degrees = 3 r1 - 4 r1^3
sin 150 degrees = 3 r2 - 4 r2^3
sin -210 degrees = 3 r3 - 4 r3^2.
sin 30 degrees = 1/2, and sin 150 = sin (180 - 150) = sin 30,
and sin -210 = sin (-210 + 360) = sin 150 = sin 30, so we see
that { r1, r2, r3 } all satisfy the cubic equation 1/2 = 3x - 4x^3,
i.e., x^3 - (3/4)x + (1/8) = 0. But the coefficients of the
polynomial with zeroes at r1, r2, r3 are (x - r1)(x - r2)(x - r3)
= x^3 - (r1 + r2 + r3)x^2 + (r1 r2 + r1 r3 + r2 r3)x - r1 r2 r3.
Identify that with x^3 - (3/4)x + (1/2) to get
r1 + r1 + r3 = 0
r1 r2 + r1 r3 + r2 r3 = -3/4
r1 r2 r3 = -1/8
Given that r1 = a = sin 10, r2 = b = sin 50, and
r3 = sin -70 = - sin 70 = -c, the first gives
a + b - c = 0, or (i) a + b = c
the third gives
ab(-c) = -1/8 or (iii) 8abc = 1
and the second and third together give
ab - ac - bc -3/4
------------ = 1/c - 1/b - 1/a = ---- = -6,
abc 1/8
or (ii) 1/a + 1/b = 1/c + 6.
Dan
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