Question

Determinati a €(0,pi) stiind ca (sin pi/7-cos a)^2+cos(pi/7-sin a)^2=2

Answer 1

[tex]\it Notez\ \ \dfrac{\pi}{7} = b,\ iar \ membrul \ st\hat{a}ng\ din\ egalitate \ devine: \\\;\\ (sinb-cosa)^2 +(cosb-sina)^2 = sin^2b -2sinbcosa +cos^2a+cos^2 b - \\\;\\ - 2sinacosb +sin^2a = (sin^2b+cos^2b)+(sin^2a+cos^2a) - \\\;\\ -2(sinbcosa+sinacosb) [/tex]

Egalitatea din enunț devine : 2 -2(sinbcosa+sinacosb) = 2 ⇒


-2(sinbcosa+sinacosb) = 0 ⇒ sinbcosa+sinacosb = 0 ⇒ sin(b+a) =0 ⇒

⇒ b+a = π ⇒ a = π - b = π - π/7 = 6π/7