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Question Number 155945 by mr W last updated on 06/Oct/21

Commented by mr W last updated on 06/Oct/21

$a c a s e w i t h t h r e e r e a l r o o t s$
	Answered by mr W last updated on 06/Oct/21

	$w e h a v e$ $\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$ $\sin^{3} θ - \frac{3}{4} \sin θ + \frac{1}{4} \sin 3 θ = 0$ $x^{3} - 3 x + 1 = 0$ $l e t x = a \sin θ, a \neq 0$ $a^{3} \sin^{3} θ - 3 a \sin θ + 1 = 0$ $\sin^{3} θ - \frac{3}{a^{2}} \sin θ + \frac{1}{a^{3}} = 0$ $\Rightarrow - \frac{3}{a^{2}} = - \frac{3}{4} \Rightarrow a = 2$ $\Rightarrow \frac{1}{a^{3}} = \frac{\sin 3 θ}{4} \Rightarrow \sin 3 θ = \frac{4}{2^{3}} = \frac{1}{2} ⩽ 1 ✓$ $\Rightarrow 3 θ = 2 k π + \sin^{- 1} \frac{1}{2} = 2 k π + \frac{π}{6}$ $\Rightarrow θ = \frac{2 k π}{3} + \frac{π}{18}$ $\Rightarrow x = a \sin θ = 2 \sin (\frac{2 k π}{3} + \frac{π}{18}), k = 0, 1, 2$ $\Rightarrow x_{1} = 2 \sin (\frac{π}{18}) \approx 0.347$ $\Rightarrow x_{2} = 2 \sin (\frac{2 π}{3} + \frac{π}{18}) = 2 \sin \frac{13 π}{18} \approx 1.532$ $\Rightarrow x_{3} = 2 \sin (\frac{4 π}{3} + \frac{π}{18}) = 2 \sin \frac{25 π}{18} \approx - 1.878$
	Commented by Tawa11 last updated on 06/Oct/21

	$Great sir .$
	Commented by ArielVyny last updated on 06/Oct/21

	$s i r c a n y o u s o l v e x^{3} - 3 x - 3 ?$
	Commented by mr W last updated on 06/Oct/21

	$f o r c a s e w i t h o n e r e a l r o o t$ $s e e Q 155928$
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