Réponse :
sachant que cos (π/5) = (1 + √5)/4
1) déterminer la valeur exacte de sin(π/5)
sin²(π/5) = 1 - cos²(π/5)
= 1 - [(1+√5)/4]²
= 1 - (1+√5)²/16
= (16 - 1 - 2√5 - 5)/16
= (10 - 2√5)/16
sin(π/5) = (√(10 - 2√5))/4
2) en déduire les cosinus et sinus de 4π/5 et de 3π/10
cos (4π/5) = cos(π - π/5) = cos(π)cos(π/5) + sin(π)sin(π/5)
= - 1 x (1+√5)/4 car sin (π) = 0
sin (4π/5) = sin(π - π/5) = sin(π)cos(π/5) - cos(π)sin(π/5)
= sin (π/5)
cos(3π/10) = cos(π/2 - π/5) = .cos(π/2)cos(π/5) + sin(π/2)sin(π/5)..= sin(π/5)
sin(3π/10) = sin(π/2 - π/5) = sin(π/2)cos(π/5) - cos(π/2)sin(π/5) = cos(π/5)
Explications étape par étape :
