[Math] Why do trigonometric equations have infinitely many solutions

Question

If i'm asked to solve the $\cos (\theta) =\frac{1}{2}$ why is the answer usually given as a general formula; in this case by: $\theta = \frac{5\pi}{3} + 2\pi k$ and $\theta = \frac{\pi}{3} + 2\pi k$; for integers $k$. Why are there infinitely many solutions?

Answer 1

In the picture below, the red line is all the points where $y=\frac12$. The green line is all the points where $y=\cos x$. Wherever the red and green lines intersect, we have both $y=\frac12$ and $y=\cos x$, so $\cos x = \frac12$.

Since the green line continues wiggling back and forth in the same manner, all the way out to infinity in both directions, and the red line continues straight to infinity in both directions, the red and green lines will intersect an infinite number of times; this shows that the equation $\cos x = \frac12$ has an infinite number of solutions.

Not all trigonometric equations have an infinite number of solutions. For example, $\cos x = 73$ has no solutions. $\cos x = kx$ has a finite number of solutions (except for $k=0$) whose number depends on $k$. For example, $\cos x = \frac1{10}x$ has 7 solutions.