Mathematics

This is a quadratic equation in disguise—treat $\cos x$ as a single variable and solve for it first. Once you find the possible values of $\cos x$, you'll need to count how many times each value actually occurs as $x$ ranges over $[0, 5\pi]$. The key is remembering that cosine oscillates between $-1$ and $1$, and over a longer interval like $[0, 5\pi]$ it completes multiple full cycles, giving you multiple solutions for each valid cosine value.

This is a quadratic equation in disguise—treat $\cos x$ as a single variable and solve for it first. Once you find the possible values of $\cos x$, you'll need to count how many times each value actually occurs as $x$ ranges over $[0, 5\pi]$. The key is remembering that cosine oscillates between $-1$ and $1$, and over a longer interval like $[0, 5\pi]$ it completes multiple full cycles, giving you multiple solutions for each valid cosine value.