Let-pi-pi-be-such-that-cos-1-and-cos-1-e-The-number-of-pairs-of-satisfying-the-above-system-of-equation-is-

Question Number 21272 by Tinkutara last updated on 18/Sep/17

Let α, β \in (- π, π) be such that

\cos (α - β) = 1 and \cos (α + β) = \frac{1}{e} .

The number of pairs of α, β satisfying

the above system of equation is

Answered by mrW1 last updated on 18/Sep/17

for α, β \in (- π, π)

α - β \in (- 2 π, 2 π)

α + β \in (- 2 π, 2 π)

\cos (α - β) = 1

\Rightarrow α - β = 0

\cos (α + β) = \frac{1}{e}

with θ = \cos^{- 1} \frac{1}{e} and 0 < θ < \frac{π}{2}

\Rightarrow α + β = \pm θ, \pm (2 π - θ)

\Rightarrow there are 4 pairs of α and β :

α = \frac{θ}{2}, β = \frac{θ}{2}

α = - \frac{θ}{2}, β = - \frac{θ}{2}

α = π - \frac{θ}{2}, β = π - \frac{θ}{2}

α = - π + \frac{θ}{2}, β = - π + \frac{θ}{2}

Commented by Tinkutara last updated on 18/Sep/17

Thank you very much Sir!