If sin3 - - sin3 -2-3-sin3 -4-3-a sin b- then the value of - ba- is

Sinθ + sin( θ+2π3)+sin( θ+4π3) =sinθ+2 sin(θ+π)cos( π3) =0 So, sin^3 θ + sin^3( θ+2π3)+sin^3( θ+4π3) =3 sinθ·sin( θ+2π3) ·sin( θ+4π3) nbsp; =3 sinθ·sin(π ...

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Byju's Answer

Standard XIII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

If sin3 θ + s...

Question

If ${sin}^{3} θ + {sin}^{3} (θ + \frac{2 π}{3}) + {sin}^{3} (θ + \frac{4 π}{3}) = a sin b θ,$ then the value of $∣ ∣ ∣ \frac{b}{a} ∣ ∣ ∣$ is

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Solution

$sin θ + sin (θ + \frac{2 π}{3}) + sin (θ + \frac{4 π}{3}) = sin θ + 2 sin (θ + π) cos (- \frac{π}{3}) = 0$
So, ${sin}^{3} θ + {sin}^{3} (θ + \frac{2 π}{3}) + {sin}^{3} (θ + \frac{4 π}{3})$
$= 3 sin θ \cdot sin (θ + \frac{2 π}{3}) \cdot sin (θ + \frac{4 π}{3})$
$= 3 sin θ \cdot sin (π - (θ + \frac{2 π}{3})) \cdot sin (π + (θ + \frac{π}{3})) = - 3 sin θ sin (\frac{π}{3} - θ) sin (\frac{π}{3} + θ) = - 3 sin θ (\frac{3}{4} - {sin}^{2} θ) = - 3 (\frac{3 sin θ - 4 {sin}^{3} θ}{4}) = \frac{- 3}{4} sin 3 θ a = \frac{- 3}{4}, b = 3 ∴ ∣ ∣ ∣ \frac{b}{a} ∣ ∣ ∣ = 4$

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If sin3θ+sin3(θ+2π3)+sin3(θ+4π3)=asinbθ, then the value of ∣∣∣ba∣∣∣ is

If ${sin}^{3} θ + {sin}^{3} (θ + \frac{2 π}{3}) + {sin}^{3} (θ + \frac{4 π}{3}) = a sin b θ,$ then the value of $∣ ∣ ∣ \frac{b}{a} ∣ ∣ ∣$ is