Graphs of $x\sin x=\sin x$ and $x=1$

Question

I put $x\sin x = \sin x$ in desmos and it shows this.

As it seems, this is not the graph of $x=1$.

My question is : Why $\sin x$ can't be cancelled from both sides of $x\sin x =\sin x$ to get $x=1$ ? The graph shows that they are not same.

Because you have to include points where sinx=0, since sinx=0 when x=kpi, k belongs to natural, all those points are solutions — Ryszard Eggink, Jul 02 '18 at 10:11

Steven Alexis Gregory · Answer 1 · 2018-07-02T19:35:12.550

3

$x \sin x = \sin x$

There are two cases to consider.

If $\sin x \ne 0$, then you can divide both sides by $\sin x$ and you get $x=1$.

If $\sin x=0$, then you get $ x = n\pi$ where $n$ can have any integer value.

So the solution set is $\{0, 1, \ \pm \pi, \ \pm 2\pi, \ \pm 3\pi, \dots \}$

edited Jul 02 '18 at 19:35

answered Jul 02 '18 at 10:11

Steven Alexis Gregory

26,769

1

What about $x=0$? – robjohn Jul 02 '18 at 14:49
@robjohn : Fixed it. Sorry about that. – Steven Alexis Gregory Jul 02 '18 at 19:35

Allawonder · Answer 2 · 2018-07-02T17:47:22.980

2

Because $\sin x$ could vanish. Your equation is equivalent to $$(\sin x)(x-1)=0,$$ which tells us that $x=1$ or $\sin x=0.$ The last equation is a warning signal that things could go wrong if we attempted indiscriminate division by $\sin x.$

PS. In general, one should be careful with taking quotients (in particular, one should bear in mind that the divisor could vanish; if the divisor is identically zero, one cannot divide).

edited Jul 02 '18 at 17:47

answered Jul 02 '18 at 13:59

Allawonder

13,327

I would say more of a hint or a signal than a warning. – robjohn Jul 02 '18 at 14:49
1

Parentheses around arguments help a lot. Out of context, it is hard to tell what is meant by $\sin x(x-1)$; is it $\sin(x(x-1))$ or $\underline{\sin(x),(x-1)}$? – robjohn Jul 02 '18 at 14:54
@robjohn Thank you. I have made some adjustments. – Allawonder Jul 02 '18 at 17:48

Graphs of $x\sin x=\sin x$ and $x=1$

2 Answers2