Question

tan 3a /tan a = k, then sin 3a/ sin a =

Answer 1

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

 

Answer 2

tan 3a / tan 4a === ULTRATAN !!!!