If-lt-lt-lt-2pi-and-cos-x-cos-x-cos-x-0-for-all-x-R-then-is-2pi-3-

Question Number 16834 by sushmitak last updated on 26/Jun/17

If α < β < γ < 2 π and

\cos (x + α) + \cos (x + β) + \cos (x + γ) = 0

for all x \in R, then is

γ - α = \frac{2 π}{3} ?

Commented by ajfour last updated on 27/Jun/17

no, i believe γ - α = \frac{4 π}{3}, then .

Commented by ajfour last updated on 27/Jun/17

Commented by prakash jain last updated on 27/Jun/17

upon expansion

\forall x

\cos (x) (\cos α + \cos β + \cos γ)

- \sin x (\sin α + \sin β + \sin γ) = 0

s i n c e a b o v e s t a t e m e n t i s t r u e

f o r a l l x

\cos α + \cos β + \cos γ = 0

\sin α + \sin β + \sin γ = 0

\cos β = - (\cos α + \cos γ) \dots \dots A

\sin β = - (\sin α + \sin γ) \dots \dots B

s q u a r e a n d a d d

1 = 2 + 2 (\cos α \cos γ + \sin α \sin γ)

\cos (γ - α) = - \frac{1}{2}

γ - α = 2 n π \pm \frac{2 π}{3}

since γ < 2 π

γ - α = \frac{2 π}{3} or \frac{4 π}{3}