How does one algebraically find the total number of intersections between $\sin(x)$ and a linear equation? Take the function $f(x) = \dfrac{x}{8}$ and $\sin(x)$. By plotting, we can see that the graphs intersect at 7 distinct points, indicated by the grey dots. However, how would one approach such a problem algebraically? There seems to be no useful way to isolate the $x$ variable in a manner that would let us solve the problem.


At $x=\dfrac{5\pi}{2}$, compare the y-values of $y=\dfrac{x}{8}$ and $y=\sin{x}$:
$\dfrac{\frac{5\pi}{2}}{8}<\dfrac{5(3.2)}{16}=\sin{\frac{5\pi}{2}}$. So there are at least $7$ intersections.
If $x=3\pi$ then $\dfrac{x}{8}=\dfrac{3\pi}{8}>1$, and $1$ is the maximum value of $\sin{x}$. So for $x\ge 3\pi$ there are no intersections.
Therefore there are exactly $7$ intersections.
