How many real solutions does the equation x=sin3x have?


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Question Number 210702 by depressiveshrek last updated on 17/Aug/24

$How many real solutions does the$ $equation x = \sin 3 x have ?$
	Answered by Frix last updated on 17/Aug/24

	$x = \sin 3 x$ $t = 3 x$ $\frac{t}{3} = \sin t$ $t_{1} = 0$ $- 1 ⩽ \frac{t}{3} ⩽ 1$ $- 3 ⩽ t ⩽ 3$ $\sin t ⩾ 0 \Rightarrow 0 ⩽ t ⩽ π; π > 3$ $\Rightarrow 1 solution for t \in (0, 3]$ $1 solution for t \in [- 3, 0)$ $\Rightarrow 3 solutions$ $\frac{t}{n} = \sin t$ $- n ⩽ t ⩽ n$ $\sin t ⩾ 0 \Rightarrow 2 k π ⩽ t ⩽ (2 k + 1) π$ $First time more than 3 solutions possible$ $at n = 7$ $- 7 ⩽ t ⩽ 7$ $0 ⩽ t ⩽ π \lor 2 π ⩽ t ⩽ 3 π$ $but in fact \frac{t}{7} > \sin t; 2 π ⩽ t$ $n = 8 \Rightarrow \frac{t}{8} = \sin t has 7 solutions$
	Answered by mr W last updated on 17/Aug/24

	$s a y 3 x = t, t h e e q u a t i o n b e c o m e s$ $\frac{t}{3} = \sin t . {l e t}^{'} s g e n e r a l l l o o k a t t h e$ $e q u a t i o n k x = \sin x .$ $w e s e e x = 0 i s a l w a y s a r o o t . b e s i d e s$ $d u e t o s y m m e t r y i f x = a i s a r o o t,$ $t h e n x = - a i s a l s o a r o o t . t h e r e f o r e$ $t h e n u m b e r o f r o o t s i s a l w a y s o d d .$ $d u e t o s y m m e t r y w e o n l y n e e d t o$ $t r e a t k ⩾ 0 . f o r x = 0, \sin x = 0 h a s$ $i n f i n i t e r o o t s . s o n o w w e t a k e k > 0 .$
	Commented by mr W last updated on 17/Aug/24

	Commented by mr W last updated on 17/Aug/24

	${(k x)}^{'} = {(\sin x)}^{'}$ $\Rightarrow k = \cos x = \sqrt{1 - k^{2} x^{2}} \Rightarrow x = \sqrt{\frac{1}{k^{2}} - 1}$ $2 (n - 1) π < x < (2 (n - 1) + \frac{1}{2}) π$ $4 {(n - 1)}^{2} π^{2} < x^{2} < 4 {(n - \frac{3}{4})}^{2} π^{2}$ $4 {(n - 1)}^{2} π^{2} + 1 < \frac{1}{k_{n}^{2}} < 4 {(n - \frac{3}{4})}^{2} π^{2} + 1$ $\frac{1}{\sqrt{4 {(n - \frac{3}{4})}^{2} π^{2} + 1}} < k_{n} < \frac{1}{\sqrt{4 {(n - 1)}^{2} π^{2} + 1}}$ $k_{n} = \cos \sqrt{\frac{1}{k_{n}^{2}} - 1}$ $t h i s e q u a t i o n h a s i n f i n i t e s o l u t i o n s :$ $k_{1}, k_{2}, k_{3}, . . .$ $k_{1} = 1$ $k_{2} \approx \frac{1}{7.789706}$ $k_{3} \approx \frac{1}{14.101695}$ $k_{4} \approx \frac{1}{20.395833}$ $. . . . . .$ $a t x = 0 : k = k_{1} = 1 .$ $i f k ⩾ k_{1} = 1, t h e r e i s o n e r o o t x = 0 .$ $i f k < k_{1} = 1, t h e r e a r e a t l e a s t 3 r o o t s .$ $g e n e r a l l y$ $f o r k = k_{n} t h e r e a r e 4 n - 3 r o o t s$ $f o r k_{n + 1} < k < k_{n} t h e r e a r e 4 n - 1 r o o t s$ $b a c k t o t h e o r i g i n a l q u e s t i o n w i t h$ $x = \sin 3 x .$ $i t i s t h e s a m e a s \frac{x}{3} = \sin x$ $s i n c e k = \frac{1}{3} < k_{1}, > k_{2} \Rightarrow i t h a s 3 r o o t s .$
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