Réponse :
sachant que cos π/5 = (√(5) + 1)/4 Déterminer les valeurs exactes de
cos (-π/5) = cos (π/5) = (√(5) + 1)/4
cos 4π/5 or 4π/5 = π - π/5 et sachant que cos (π - x) = - cos x
donc cos 4π/5 = cos (π - π/5) = - cos π/5 = - (√(5) + 1)/4 = (
sin π/5 on utilise la relation sin² x + cos² x = 1 ⇔ sin² x = 1 - cos² x
sin x = √(1 - cos² x)
donc sin π/5 = √(1 - cos² π/5)
or cos π/5 = (√(5) + 1)/4 ⇒ cos² π/5 = [(√(5) + 1)/4]² = (5 + 2√5 + 1)/16
= (6 + 2√5)/16
= 2(3 + √5)/16
= (3+√5)/8
et 1 - (3+√5)/8 = (8 - 3 - √5)/8 = (5 - √5)/8
donc sin π/5 = √((5 -√5)/8) = √(5 - √5)/√8 = √(5 - √5)/2√2
= √2√(5 - √5)/4
= (√2(5 - √5))/4
= (√(10 - 2√5))/4
Explications étape par étape
