How many solutions does $\cos(97x)=x$ have?

I have plot the function. However I don't know how to solve the problem without computer.

Can anyone give a fast solution without a computer?

  • 2
    $\begingroup$ It's a GRE Problem. $\endgroup$
    – little o
    Oct 19, 2018 at 6:08

2 Answers 2


Let us concentrate on $[0,1]$ as suggested by Siminore. The period of $\cos(97x)$ is $\frac{2\pi}{97}$. On $[0,1]$, the function repeats itself $$ \frac{1}{\frac{2\pi}{97}}\approx 15.43 $$ On each period, the functions $\cos(97x),x$ meet twice, so you get at least $30$ meetings. Since it does almost half a period after its $15^{th}$ before going over $1$, you can convince yourself that on the $0.43$ part, they will meet again, so we have $31$ meetings.

On $[-1,0]$ the function will also repeat approximatively $15.43$ times, however, this time, it won't be enough to get another meeting, so there will only be $30$ on that part. There is then a total of $61$ solutions.

For clarity, they meet $30$ times in $[0,15\times\frac{2\pi}{97}]$ and since $15\times\frac{2\pi}{97}\approx 0.97<1$ and $\cos(97\times 1)\approx -0.925 $, the functions will meet once more in $[15\times\frac{2\pi}{97},1]$.

On the other hand, after the $15^{th}$ period of $\cos(97x)$ (Imagine it starts from $0$ and goes backward to $-1$), $x$ is about $-0.97$ and decreasing, while $cos(97x)$ is at $1$ decreasing up to $\cos(-97)\approx -0.925$, not enough for another meeting

  • $\begingroup$ But this is not symmetric, and there is actually one less solution on the $x<0$ side. $\endgroup$
    – Jonathan
    Nov 3, 2012 at 16:11
  • $\begingroup$ @Jonathan you are right, I should have plotted both side $\endgroup$ Nov 3, 2012 at 16:14

Since $-1 \leq \cos (97x) \leq 1$, you can restrict your attention to the interval $[-1,1]$, and, even better, to $[0,1]$ by evenness.

If you sketch the graph of $x \mapsto \cos (97x)$, you'll understand that you need to count the "bumps" of this function lying in the upper half-plane, and add 1 since the first "bump" is crossed by the $y$-axis. See the graph here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.