If cos -2 cos-3 cos-sin-2 sin-3 -0- then sin 3 -8 sin 3 -27 sin 3 -

The correct option is B Let x = cosθ + i sinθ, y = 2(cosϕ + isinϕ) nbsp; nbsp; nbsp; z = 3(cosΨ + isinΨ) Then x + y + z = (cosθ + 2cosϕ + 3cosψ) + i( ...

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Standard XI

Mathematics

Purely Real

If cos θ+2 co...

Question

If $c o s θ + 2 c o s ϕ + 3 c o s Ψ = s i n θ + 2 s i n ϕ + 3 Ψ = 0,$ then $s i n 3 θ + 8 s i n 3 ϕ + 27 s i n 3 ψ =$

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Solution

The correct option is C

Let x = $c o s θ + i s i n θ, y = 2 (c o s ϕ + i s i n ϕ)$

z = 3 $(c o s Ψ + i s i n Ψ)$

Then x + y + z = $(c o s θ + 2 c o s ϕ + 3 c o s ψ) + i (s i n θ + 2 s i n ϕ + 3 s i n ψ)$

= 0

$∴$ $x^{3} + y^{3} + z^{3} = 3 x y z$

Thus $(c o s 3 θ + i s i n 3 θ) + 8 (c o s ϕ + i s i n 3 ϕ) + 27 (c o s 3 Ψ + i s i n 3 Ψ)$

= 3.2.3 . ${c o s (θ + ϕ + Ψ) + i s i n (θ + ϕ + Ψ)}$ .

Equation imaginary parts, we get

$s i n 3 θ + 8 s i n 3 Φ + 27 s i n 3 Ψ = 18 s i n (θ + ϕ + Ψ)$ .

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If cosθ+2cosϕ+3cosΨ=sinθ+2sinϕ+3Ψ=0, then sin3θ+8sin3ϕ+27sin3ψ=

If $c o s θ + 2 c o s ϕ + 3 c o s Ψ = s i n θ + 2 s i n ϕ + 3 Ψ = 0,$ then $s i n 3 θ + 8 s i n 3 ϕ + 27 s i n 3 ψ =$