Question

Resolva a seguinte equação trigonométrica para x ∈ [0, 2π]:

sen⁶(x/2) + cos⁶(x/2) = 7/16

Answer 1

Boa noite

sen⁶(x/2) + cos⁶(x/2) = 7/16

identidade trigonometria

1 -  3sen²(x/2) + 3sen⁴(x/2) =  7/16

mudança de variável 

y = sen²(x/2)

9/16 - 3y + 3y² = 0

delta
d² =  9 - 4*9/16*3
d² = 9 - 27/4 = 36/4 - 27/4
d² = 9/4
d = 3/2

y1 =  (3 + 3/2)/6 = 3/4
y2 = (3 - 3/2)/6 = 1/4

y = sen²(x/2)

sen²(x/2) = 3/4
sen²(x/2) = 1/4 

sen(x/2) = +√3/2 , x = 4π/3, x = 2π/3 
sen(x/2) = -√3/2  , x = 8π/3, x = 4π/3
sen(x/2) = +1/2   , x = π/3,  x = 5π/3 
sen(x/2) = -1/2    , x = 11π/3, x = 19π/3

S = {π/3, 2π/3, 4π/3, 5π/3}.