Solving equations involving $x, \pi, \cos x$.

Question

How are such equations solved?

Question 1. Find the number of roots of the equation $2x=3\pi (1-\cos x)$ where $x$ is measured in radians.

Question 2. Find the number of real roots of the equation $2x=(2n+1)\pi (1-\cos x)$.

Answer 1

Use a plotting program to plot the two functions $$ y = 2x \\ y = 3\pi(1 - \cos x). $$ Where the two graphs intersect, you get a root of your (first) equation:

You will see (and you can easily verify) that there is a root at $x=0$ and another root at $x = 3\pi$. What's not quite obvious is that there are two more intersections very close to those two. If your plot is coarse (as the one above is), it might appear that the straight line graph of the first function is tangent to the graph of the second function, but that's not the case (and in situations like that, where you almost have a double root, you need to take special care with any numerical attempt to determine the roots). We can zoom in to take a closer look:

You can argue as follows to convince yourself that there are two roots near each point: at those two points, the second function has a minimum or a maximum, so the derivative is zero. OTOH, the derivative of the first function is 2, so the graph of the first function must intersect the graph of the second function at a second point near each of the solutions above: a little to the right of $x=0$ and a little to the left of $x=3\pi$. BTW, this is not a proof, just a plausibility argument, but with some effort it can be made into a proper proof.

Finally, there is another root, right in the middle between the outer roots at $x=\frac{3}{2}\pi$.

You should now be able to repeat the process for $n=2$ and then generalize to any $n$.