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$Q:$ How many distinct values of x satisfy the equation $sinx=\dfrac{x}{3}$ where x is in radians.
in Calculus by (155 points) | 41 views

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$Ans: 3$


1. $At\ x=0 \sin x=0$ & $\frac{x}{3}=0 $

2. $At\ x=\frac{\Pi}{2},\ \sin x=1\ but\ \frac{x}{3}=\frac{\pi}{6}\ i.e\ \sin x > \frac{x}{3} $ &

$At\ x=\Pi,\ \sin x=0\ but\ \frac{x}{3}=\frac{\pi}{3}\ i.e\ \sin x < \frac{x}{3} $

So according to $Intermediate\ Value\ Theorem$ there exists atleast one solution in the range $\left[\frac{\Pi}{2},\pi \right]$

And from $x=\frac{\Pi}{2}\ to\ x=\pi,\ \sin x$ is decreasing and $\frac{x}{3}$ is increasing, so there will be only one solution in the range $\left[\frac{\Pi}{2},\pi \right]$

3. Similarly one solution exists in the range $\left[-\frac{\Pi}{2},-\pi \right]$

P.S : $Intermediate\ Value\ Theorem$ : $Suppose\ f\ is\ function\ continuous\ at\ every\ point\ of\ the\ interval\ [a,b]\ :\ $

  • $f\ will\ take\ on\ every\ value\ b/w\ f(a)\ and\ f(b)\ over\ the\ interval$

  • $for\ any\ L\ b/w\ the\ values\ f(a)\ and\ f(b),\ there\ exist\ a\ number\ c\ in\ [a,b]\ for\ which\ f(c)=L\ $

by (666 points)
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@pranay562

u mean 2.279 radians?? then how much in degree??

0
Yes 2.279 radians. It equals to 130.5°. To convert radians to degrees just multiply the radians with $\frac{180}{\pi}$.
0

@pranay562

I understood, what u mean.

U mean, there are two equations

$y=\sin x$

and   $y=\frac{x}{3}$

rt?

@Kushagra गुप्ता

what is ans given?

0
yes ma'am and answer is 3
0

@srestha ma'am,

answer is given as 3

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