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I plotted the graphs of sin8x=y and y=x and I know that they intersect each other in seven points but I want to know is there any other solution without plotting them. Thanks a lot.

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    Do you only want to find out about the *number* of the points or also their (more or less) exact *positoin*? – 2017-02-28
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    Hi. Yes. I want to know the number of points which their intersect each other. – 2017-02-28

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As both functions are odd, the graphs clearly intersect at $(0,0)$, and we need only count the number of intersections for positive $x$ (then double this count and add one). Also, the sine is at most $1$ in absolute value, hence we need only check up to $x=1$. Incidentally, $8$ is slightly above $\frac 52\pi$, hence at $x=\frac \pi{16}, \frac {5\pi}{16}$ we have $\sin 8x=1>x$, whereas at $x=\frac{3\pi}{16}$, we have $\sin 8x=-1at least one intersection point $\in(\frac{\pi}{16},\frac{3\pi}{16})$, one $\in(\frac{3\pi}{16},\frac{5\pi}{16})$, and one $\in(\frac{5\pi}{16},1)$.

Can there be more? Between any two points (or at any multiple intersection), the mean value theorem demands a point where $8\cos 8x = 1$. This equation certainly has exactly one solution $\in(0,\frac\pi8)$, one $\in(\frac\pi 8,\frac{\pi}4)$, one $\in(\frac\pi4,1)$.

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    I plotted sinx=y and x=y in Matlab, and I found that they intersect each other in seven points. – 2017-02-28
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I mean, I want to plot y=x and y=sin8x. my question is in which points they intersect each other? y=x is Bisect the first and third quarter.