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 :