Intersection of a trigonometric function and a line

An intersection of a trigonometric function and a line amounts to solving a trigonometric equation. We find where $y = \sin x$ meets the line $y = \dfrac{1}{2}$.

At an intersection $\sin x = \dfrac{1}{2}$. On $0 \leq x < 2\pi$, this holds at $x = \dfrac{\pi}{6}$ and $x = \dfrac{5\pi}{6}$. Sine takes the value $\dfrac{1}{2}$ at $x = \dfrac{\pi}{6}$ and at the mirror position $x = \pi - \dfrac{\pi}{6} = \dfrac{5\pi}{6}$.

But $\sin x$ repeats with period $2\pi$, so these are not the only intersections. The same crossings recur every $2\pi$, and the solutions are $x = \dfrac{\pi}{6} + 2\pi n$ and $x = \dfrac{5\pi}{6} + 2\pi n$ (with $n$ an integer) — infinitely many.

Sliding the line changes the number of intersections. A line $y = k$ with $-1 < k < 1$ gives two per period; $k = 1$ or $k = -1$ gives one per period (tangent at a peak or trough); and $|k| > 1$ gives none. The large dots are the two intersections on $0 \leq x < 2\pi$: $\left(\dfrac{\pi}{6}, \dfrac{1}{2}\right)$ and $\left(\dfrac{5\pi}{6}, \dfrac{1}{2}\right)$.