If cos α + 2 cos β + 3 cos γ = 0, sin α + 2 sin β + 3 sin γ = 0 and α + β + γ = π, then sin 3α + 8 sin 3β + 27 sin 3γ =

Question

If cos α + 2 cos β + 3 cos &gamma; = 0, sin α + 2 sin β + 3 sin &gamma; = 0 and α + β + &gamma; = π, then sin 3α + 8 sin 3β + 27 sin 3&gamma; = (A) -18 (B) 0 (C) 3 (D) 9

Tardigrade Admin · Accepted Answer

Let a=cos α + i sin α b = cos β + i sin β and c = cos &gamma; + i sin &gamma; Then, a + 2b + 3c = ( cos α + 2 cos β + 3 cos &gamma;) + i (sin α + 2 sin β + 3 sin &gamma;)= 0 ⇒ a3 + 8b3 + 27c3 = 18abc (∵ if x + y + z = 0 ⇒ x3 + y3 + z3 = 3xyz) ⇒ cos 3 α + 8 cos 3 β + 27 cos 3&gamma; = 18 cos (α + β + &gamma;) and sin 3 α + 8 sin 3 β + 27 sin 3&gamma; = 18 sin (α + β + &gamma;) = 18 sin π = 0

Q. If cosα+2cosβ+3cosγ=0,sinα+2sinβ+3sinγ=0 and α+β+γ=π, then sin3α+8sin3β+27sin3γ =

-18

0

3

9

Let a=cosα+isinα b=cosβ+isinβ and c=cosγ+isinγ Then, a+2b+3c=(cosα+2cosβ+3cosγ)+i(sinα+2sinβ+3sinγ)=0 ⇒a3+8b3+27c3=18abc (∵ifx+y+z=0 ⇒x3+y3+z3=3xyz) ⇒cos3α+8cos3β+27cos3γ =18cos(α+β+γ) and sin3α+8sin3β+27sin3γ =18sin(α+β+γ) =18sinπ=0

Q. If $cos α + 2 cos β + 3 cos γ = 0, sin α + 2 sin β + 3 sin γ = 0$ and $α + β + γ = π,$ then $sin 3 α + 8 sin 3 β + 27 sin 3 γ$ =