sin^2 x+sinx=(4/3) , x∈[0,π] equation sum of roots in range?


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Question Number 141110 by Fikret last updated on 15/May/21

${s i n}^{2} x + s i n x = \frac{4}{3}, x \in [0, π]$ $e q u a t i o n s u m o f r o o t s i n r a n g e ?$
	Answered by MJS_new last updated on 15/May/21

	$s^{2} + s - \frac{4}{3} = 0 \land - 1 ⩽ s ⩽ 1 \Rightarrow s = - \frac{1}{2} + \frac{\sqrt{57}}{6}$ $\sin x = - \frac{1}{2} + \frac{\sqrt{57}}{6} \land 0 ⩽ x ⩽ π \Rightarrow$ $x_{1} = \arcsin (- \frac{1}{2} + \frac{\sqrt{57}}{6})$ $x_{2} = π - \arcsin (- \frac{1}{2} + \frac{\sqrt{57}}{6})$ $x_{1} + x_{2} = π$
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