0
$\begingroup$

How many solutions are there to the equation $\cos(31x) = \sqrt{3}/2$ on the interval $[-4\pi,2\pi)$? Justify your answer.

Since $\pi/6$ and $11\pi/6$ yields $\sqrt{3}/2$ in function $\cos(x)$, I did $31x = \pi/6+2n\pi$ and the same for the other and put both of them in $[-4\pi,2\pi)$ to solve the inequality. In the end, I got that (since n has to be a positive integer) $n=30$. However, the answer is supposed to be 186. How can I solve this?

| cite | improve this question | | | | |
$\endgroup$
  • $\begingroup$ Note that $n$ can also be a negative integer. Also, the fact that $x \in [-4\pi, 2\pi)$ implies that $31x \in [-124\pi, 62\pi)$. Did you take this into account when counting the number of possible $n$'s? $\endgroup$ – levap Aug 13 '17 at 8:41
1
$\begingroup$

Solving $\cos(31x) = \frac{\sqrt{3}}{2}$ on $[-4\pi, 2\pi)$ is equivalent to solving $\cos(u) = \frac{\sqrt{3}}{2}$ on $[-124 \pi, 62 \pi)$. The reason is that as $x$ ranges in $[-4\pi, 2\pi)$, the variable $u = 31x$ ranges in $$[31 \cdot (-4\pi), 31 \cdot (2\pi)) = [-124\pi, 62 \pi).$$

On each interval of the form $[\pi k, \pi(k + 1))$ (where $k \in \mathbb{Z}$), the equation $\cos(u) = \frac{\sqrt{3}}{2}$ has only one solution. Hence, the total number of solutions is $124 + 62 = 186$.

| cite | improve this answer | | | | |
$\endgroup$

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.