Intersection of circle and sine wave

I recently encountered the following question [University of Tor Vergata, Engineering Sciences, PreCalculus self assessment test, Geometry D]: A circle has center at the point A = (1, 1) and has radius r = 2. At how many points does it intersect the function y = sin(x)?

The equation of the circle with center (1, 1) and radius 2 is: (x - 1)² + (y - 1)² = 4

At any intersection point, the coordinates (x, y) must satisfy both equations:

1. (x - 1)² + (y - 1)² = 4

2. y = sin(x)

Let's substitute the second equation into the first: (x - 1)² + (sin(x) - 1)² = 4

This is a transcendental equation and cannot be solved analytically. Luckily, the equations are easy to plot by hand. The circle is trivial and plotting sin(x) only requires you to know that sin(0)=0, sin(pi/2)=1, sin(2*pi)=0. The sine wave amplitude is between -1 and 1. Since the circle has radius 2 and is centered at (1, 1), both the x and y interval for the circle will be [-1, 3]. This results in two intersections. Using Desmos for a cleaner plot:

Note that if we increase the amplitude of the sine wave, we will can have more than 2 intersections. For an amplitude of 3.5, we have 4 intersections:

After amplitude, if we also increase the frequency by 4, we get even more intersections:

It is only reasonable to find the number of intersections by hand drawing if the sine wave is of the simple form sin(x).

If you want to find the numerical values of the intersections (which is not asked for in the question), you have to use numerical root finding methods like Newton-Raphson.