Find the values of x in the range 0° to 360° for which sin(3x)sin(x) = 2cos(2x) + 1


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Question Number 13903 by tawa tawa last updated on 24/May/17

$Find the values of x in the range 0 ° to 360 ° for which$ $\sin (3 x) \sin (x) = 2 \cos (2 x) + 1$
Commented by myintkhaing last updated on 26/May/17

$P l e a s e i n t h e r a n g e 0 ° t o 360 ° m e a n s$ $0 ° < x < 360 ° o r 0 ° ⩽ x ⩽ 360 ° ? ?$
	Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

	$(3 s i n x - 4 {s i n}^{3} x) s i n x = 2 (1 - 2 {s i n}^{2} x) + 1$ $s i n x = t \Rightarrow 3 t^{2} - 4 t^{4} = - 4 t^{2} + 3 \Rightarrow$ $4 t^{4} - 7 t^{2} + 3 = 0 \Rightarrow t^{2} = 1, \frac{3}{4} \Rightarrow t = s i n x = \pm 1, \pm \frac{\sqrt{3}}{2}$ $s i n x = 1 \Rightarrow x = 2 k π + \frac{π}{2}$ $s i n x = - 1 \Rightarrow x = 2 k π + \frac{3 π}{2}$ $s i n x = \frac{\sqrt{3}}{2} \Rightarrow x = 2 k π + \frac{π}{3}, 2 k π + \frac{2 π}{3}$ $s i n x = - \frac{\sqrt{3}}{2} \Rightarrow x = 2 k π + \frac{4 π}{3}, 2 k π + \frac{5 π}{3}$ $\Rightarrow x = \frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3} \in (0, 2 π) . ◼$
	Commented by tawa tawa last updated on 25/May/17

	$God bless you sir .$
	Commented by ajfour last updated on 25/May/17

	$i t h i n k x = \frac{2 π}{3}, \frac{5 π}{3} a l s o q u a l i f y$ $i n t h e g i v e n r a n g e .$
	Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

	$y o u a r e r i g h t s i r . t h a n k s .$
	Answered by ajfour last updated on 25/May/17

	$2 \sin (3 x) \sin (x) = 4 \cos (2 x) + 2$ $\cos (2 x) - \cos (4 x) = 4 \cos (2 x) + 2$ $\cos (4 x) + 3 \cos (2 x) + 2 = 0$ $2 \cos^{2} (2 x) - 1 + 3 \cos (2 x) + 2 = 0$ $l e t t = \cos (2 x), t h e n$ $2 t^{2} + 3 t + 1 = 0$ $(2 t + 1) (t + 1) = 0$ $t = - \frac{1}{2}; t = - 1$ $\Rightarrow \cos (2 x) = - \frac{1}{2} w h e r e 0 ⩽ 2 x ⩽ 4 π$ $2 x = π \pm \frac{π}{3}, 3 π \pm \frac{π}{3}$ $x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$ $a n d i f \cos (2 x) = - 1$ $2 x = π, 3 π$ $x = \frac{π}{2}, \frac{3 π}{2}$ $h e n c e i f 0 ⩽ x ⩽ 2 π, t h e n$ $x = \frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3} .$
	Commented by mrW1 last updated on 25/May/17

	$y e s, t h e r e a r e 6 s o l u t i o n s i n t h e$ $r a n g e (0, 2 π)$
	Commented by ajfour last updated on 25/May/17

	$t h a n k s .$
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