If cos-2 cos-3 cos-sin-2 sin-3 sin-0- then the value of sin 3 -8 sin 3 -27 sin 3 Y isA- sin -B- 3 sin - y - 18 sin - y D- sin -2 -3 y

Let a=cosα+isinα, b=cosβ+isinβ, c=cosγ+isinγ Then, a+2b+3c=(cosα+2cosβ+3 cosγ)+i(sinα+2sinβ+3 sinγ)=0 ⇒ a^3+8b^3+27c^3=18abc ⇒cos 3α+8cos 3β+27 cos 3γ nbsp;=1 ...

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Standard XII

Mathematics

Polar Representation of a Complex Number

If cosα+2 cos...

Question

If $cos α + 2 cos β + 3 cos γ = sin α + 2 sin β + 3 sin γ = 0$ , then the value of $sin 3 α + 8 sin 3 β + 27 sin 3 γ$ is

sin (α + β + γ)

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3 sin (α + β + γ)

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18 sin (α + β + γ)

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sin (α + 2 β + 3 γ)

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Solution

The correct option is C $18 sin (α + β + γ)$
Let $a = cos α + i sin α, b = cos β + i sin β, c = cos γ + i sin γ$
Then,
$a + 2 b + 3 c = (cos α + 2 cos β + 3 cos γ) + i (sin α + 2 sin β + 3 sin γ) = 0$
$\Rightarrow a^{3} + 8 b^{3} + 27 c^{3} = 18 a b c$
$\Rightarrow cos 3 α + 8 cos 3 β + 27 cos 3 γ = 18 cos (α + β + γ)$
and $sin 3 α + 8 sin 3 β + 27 sin 3 γ = 18 sin (α + β + γ)$

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Similar questions

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If cosα+2cosβ+3cosγ=sinα+2sinβ+3sinγ=0, then the value of sin3α+8sin3β+27sin3γ is

The correct option is C 18sin(α+β+γ)Let a=cosα+isinα, b=cosβ+isinβ, c=cosγ+isinγ Then, a+2b+3c=(cosα+2cosβ+3cosγ)+i(sinα+2sinβ+3sinγ)=0 ⇒a3+8b3+27c3=18abc ⇒cos3α+8cos3β+27cos3γ=18cos(α+β+γ) and sin3α+8sin3β+27sin3γ=18sin(α+β+γ)

If $cos α + 2 cos β + 3 cos γ = sin α + 2 sin β + 3 sin γ = 0$ , then the value of $sin 3 α + 8 sin 3 β + 27 sin 3 γ$ is