cos²a + cos²(a+120°) + cos(a-120°) = 3/2

Answer 1

after some of those others you posted, this one is pretty mechanical
cos 120° = -1/2
sin 120° = √3/2

(-1/2)^2 + (cosa * (-1/2) - sina*(√3/2))^2 + (cosa * (-1/2) + sina*(√3/2))^2

1/4 + 1/4 (cos^2a + 2√3 sina*cosa + 3sin^2a) + 1/4 (cos^2a - 2√3 sina*cosa + 3sin^2a)

1/4 (1 + 2cos^2a + 6sin^2a)
1/4 (1 + 2 + 4sin^2a)

3/4 + sin^2a = 3/2
sin^2a = 3/4
sina = ±√3/2

a = pi/3, 2pi/3, 4pi/3, 5pi/3