find solution set of equation cos 3x = −cos x , x∈(0,3π)


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Question Number 114272 by bemath last updated on 18/Sep/20

$f i n d s o l u t i o n s e t o f e q u a t i o n$ $\cos 3 x = - \cos x, x \in (0, 3 π)$
Commented by malwaan last updated on 18/Sep/20

$(i) c o s 3 x = c o s (π - x)$ $\Rightarrow 3 x = \pm (π - x) + 2 k π$ $(a) 3 x = (π - x) + 2 k π$ $\Rightarrow x = π (\frac{1 + 2 k}{4})$ $(b) 3 x = - (π - x) + 2 k π$ $\Rightarrow x = π (\frac{2 k - 1}{2})$ $(i i) c o s 3 x = c o s (π + x)$ $\Rightarrow 3 x = \pm (π + x) + 2 k π$ $(a) 3 x = (π + x) + 2 k π$ $\Rightarrow x = π (\frac{1 + 2 k}{2})$ $(b) 3 x = - (π + x) + 2 k π$ $\Rightarrow x = π (\frac{2 k - 1}{4})$ $∴ s o l u t i o n s e t =$ ${π (\frac{2 k \pm 1}{2}); π (\frac{2 k \pm 1}{4})}$
Commented by bemath last updated on 18/Sep/20

$h a h a h a s a n t u y$
Commented by abdullahquwatan last updated on 18/Sep/20

$k e n a p a p a k e s a n t u y s i h ? ?$
	Answered by bemath last updated on 18/Sep/20

	Answered by malwaan last updated on 18/Sep/20

	$x = {π (\frac{2 k \pm 1}{2}); π (\frac{2 k \pm 1}{4})}$ $x \in (0, 3)$ $k = 1 \Rightarrow x = \frac{π}{2}; \frac{3 π}{2}; \frac{π}{4}; \frac{3 π}{4} (< 3 π)$ $k = 2 \Rightarrow x = \frac{5 π}{2}; \frac{5 π}{4} (< 3 π)$ $k = 3 \Rightarrow x = \frac{7 π}{4} (< 3 π)$ $k = 4 \Rightarrow x = \frac{9 π}{4} (< 3 π)$ $k = 5 \Rightarrow x = \frac{11 π}{4} (3 π)$
	Answered by Dwaipayan Shikari last updated on 18/Sep/20

	$c o s 3 x + c o s x = 0$ $2 c o s 2 x c o s x = 0$ $c o s 2 x = 0$ $2 x = k π + \frac{π}{2}$ $x = \frac{k π}{2} + \frac{π}{4}$ $o r$ $x = k π + \frac{π}{2} (k \in Z)$ $S o l u t i o n s e t$ $x \in [\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}, \frac{9 π}{4}, \frac{11 π}{4}]$ $x \in [\frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}]$
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