tan(3a)/tan(a) = sin(3a)/sin(a) * cos(a)/cos(3a)
using De Moivre's formula
cos(3a)+isin(3a) = (cos(a)+isin(a))³
then cos(3a) = cos³(a)-3cos(a)sin²(a) = cos(a) [1-4sin²(a)]
and sin(3a) = 3cos²(a)sin(a)-sin³(a) = sin(a) [3-4sin²(a)]
thus cos(3a)/cos(a) = 1-4sin²(a) = sin(3a)/sin(a) -2
let x = sin(3a)/sin(a)
then tan(3a)/tan(a) = k = x / (x-2)
kx - 2k = x
x(k - 1) = 2k
x = sin(3a)/sin(a) = 2k / (k-1)
only if a different of :
-
h*pi/2
-
pi/6 + h/pi/3
-
- pi/6 + h*pi/3 with h in Z