Trigonometry

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

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  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

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