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Properties of nth Root of a Complex Number

If cosα +2cos...

Question

If $cos α + 2 cos β + 3 cos γ = sin α + 2 sin β + 3 sin γ = 0,$ then

A

cos 3 α + 8 cos 3 β + 27 cos 3 γ = 18 cos (α + β + γ)

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B

cos 3 α + 8 cos 3 β + 27 cos 3 γ = 18 sin (α + β + γ)

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C

sin 3 α + 8 sin 3 β + 27 sin 3 γ = 18 sin (α + β + γ)

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D

sin 3 α + 8 sin 3 β + 27 sin 3 γ = 18 cos (α + β + γ)

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Solution

The correct option is C $sin 3 α + 8 sin 3 β + 27 sin 3 γ = 18 sin (α + β + γ)$
Given,
$cos α + 2 cos β + 3 cos γ = 0 sin α + 2 sin β + 3 sin γ = 0$

Let, $a = cos α + i sin α = e^{i α}$
$b = cos β + i sin β = e^{i β}$
$c = cos γ + i sin γ = e^{i γ}$

Now,
$a + 2 b + 3 c = (cos α + 2 cos β + 3 cos γ) + i (sin α + 2 sin β + 3 sin γ)$
$\Rightarrow a + 2 b + 3 c = 0$
${If A + B + C = 0, then A^{3} + B^{3} + C^{3} = 3 A B C}$

$∴ a^{3} + 8 b^{3} + 27 c^{3} = 18 a b c$
$\Rightarrow e^{3 i α} + 8 e^{3 i β} + 27 e^{3 i γ} = 18 e^{i (α + β + γ)}$
By equating real and imaginary parts, we have
$cos 3 α + 8 cos 3 β + 27 cos 3 γ = 18 cos (α + β + γ)$
and, $sin 3 α + 8 sin 3 β + 27 sin 3 γ = 18 sin (α + β + γ)$

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Properties of nth Root of a Complex Number

Standard XI Mathematics

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