2(√3) sin^2 (x+((3π)/2)) + sin 2x = 0 0≤x≤2π


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Question Number 105112 by bemath last updated on 26/Jul/20

$2 \sqrt{3} \sin^{2} (x + \frac{3 π}{2}) + \sin 2 x = 0$ $0 ⩽ x ⩽ 2 π$
	Answered by john santu last updated on 28/Jul/20
	$⇔ 2(√3) cos^2 (x)+ sin (2x) = 0 2cos (x) {(√3) cos x+sin x } = 0 { ((cos x = 0 →x = (π/2)+2kπ)),((tan x = −(√3) = tan ((2π)/3) →x=((2π)/3)+kπ)) :} { ((x = (π/2))),((x = ((2π)/3) ; ((5π)/3))) :} (JS ♠★)$
	$\Leftrightarrow 2 \sqrt{3} \cos^{2} (x) + \sin (2 x) = 0$ $2 \cos (x) {\sqrt{3} \cos x + \sin x} = 0$ ${\begin{cases} \cos x = 0 \to x = \frac{π}{2} + 2 k π \\ \tan x = - \sqrt{3} = \tan \frac{2 π}{3} \to x = \frac{2 π}{3} + k π \end{cases}$ ${\begin{cases} x = \frac{π}{2} \\ x = \frac{2 π}{3}; \frac{5 π}{3} \end{cases} (J S ♠ ★)$
	Answered by Dwaipayan Shikari last updated on 26/Jul/20
	$2(√3)cos^2 x+2sinxcosx=0 4cosx(((√3)/2)cosx+(1/2)sinx)=0 4cosx=0 x=(4k+1)(π/2)......(a) or(((√3)/2)cosx+(1/2)sinx)=0 sin((π/3)+x)=0 (π/3)+x=kπ x=kπ±(π/3) { ((x=(π/2))),((x=((2π)/3),((5π)/3))) :}$
	$2 \sqrt{3} {c o s}^{2} x + 2 s i n x c o s x = 0$ $4 c o s x (\frac{\sqrt{3}}{2} c o s x + \frac{1}{2} s i n x) = 0$ $4 c o s x = 0$ $x = (4 k + 1) \frac{π}{2} . . . . . . (a)$ $o r (\frac{\sqrt{3}}{2} c o s x + \frac{1}{2} s i n x) = 0$ $s i n (\frac{π}{3} + x) = 0$ $\frac{π}{3} + x = k π$ $x = k π \pm \frac{π}{3}$ ${\begin{cases} x = \frac{π}{2} \\ x = \frac{2 π}{3}, \frac{5 π}{3} \end{cases}$
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