Find the range of x such that { ((sinx>0)),(((√3)sinx+cosx>0)),((0<x<2π)) :}


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Question Number 181521 by CrispyXYZ last updated on 26/Nov/22

$Find the range of x such that$ ${\begin{cases} \sin x > 0 \\ \sqrt{3} \sin x + \cos x > 0 \\ 0 < x < 2 π \end{cases}$
	Answered by mr W last updated on 26/Nov/22

	$\sin x > 0 \Rightarrow 0 < x < π$ $\sqrt{3} \sin x + \cos x > 0 \Rightarrow \cot x > - \sqrt{3}$ $\Rightarrow 0 < x < \frac{5 π}{6}$
	Answered by mahdipoor last updated on 26/Nov/22

	$\sqrt{3} s i n x + c o s x = 2 (\frac{\sqrt{3}}{2} s i n x + \frac{1}{2} c o s x) =$ $2 (c o s 30 s i n x + s i n 30 c o s x) = 2 s i n (30 + x) > 0$ $\Rightarrow 360 n < 30 + x < 360 n + 180$ $\Rightarrow s i n x > 0 \Rightarrow 360 n < x < 360 n + 180$ $\Rightarrow 0 < x < 360$ $\Rightarrow\Rightarrow\Rightarrow 0 < x < 150 o r 0 < x < \frac{5}{6} π$
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