# The root of a trigonometric equation

I want to find the least $3>p>\sqrt{5}$ such that $$f_p(x)=(p x+x) \sin \left(x-\frac{x}{p}\right)+(p x-x)\sin \left(\frac{x}{p}+x\right)-2 p \cos\left(\frac{2 x}{p}\right)+2 p \cos\left(x-\frac{x}{p}\right)+2 p \cos\left(\frac{x}{p}+x\right)-2 p=0$$ has a solutions $x\in\left(0,\frac{\pi}{2}\right)$. I use Mathematica to plot $f_p(x)$ for various p. It seems there exists a critical value $p_0$ between $2.24$ and $2.25$ such that if $3>p>p_0$, then $f_p(x)$ has a solution $x\in\left(0,\frac{\pi}{2}\right)$.However, I have no idea to give a rigruous proof and determine $p_0$. Any suggestion, idea, or comment is welcome, thanks!

(Not a rigorous proof)

If you plot $f_p(x)$ over various $p$ you will find that the solution $x$ is increasing with $p$. Everything follows depends on this observation. The minimal $p$ which a solution exists in $x\in(0,\frac\pi2)$ is $p=\sqrt5\approx2.23607$, which could be found by solving $f^{(4)}_p(0)=-\frac{4}{p^3}(p^4-6p^2+5)=0$ (all lower derivatives are identically zero; we are trying to show $x=0$ changes from a quadruple root to quintuple root).

Although this is what you asked, I doubt this is what you really want.

The maximal $p$ can be found by solving $$0 = f_p\left(\frac\pi2\right) = p\left( \pi \cos \frac{\pi}{2p} - 2\left(1 + \cos\frac\pi p\right) \right) \implies \cos\frac{\pi}{2p} = \frac\pi4,$$ i.e. $p = \frac{\pi}{2\cos^{-1}(\pi/4)} \approx 2.35340$.

The range of $p$ where solution exists between 0 and $\frac\pi2$ is $2.23607 < p < 2.35340$. Between 2.35340 and 3, there are no solutions $x\in(0,\frac\pi2)$.

• Thank you for the detailed calculation. Using Mathematica (how did you plot the figure? Mathematica?), I didn't find the solution $x$ is increasing with $p4. So now the question is, how to show this fact rigorously. Do you have any idea? – LCH Mar 20 '17 at 17:15 • Yes, you are right. The maximal$p$is also what I want. – LCH Mar 20 '17 at 17:17 • @LCH (1) ContourPlot (2) Maybe you could argue by (a) there is a unique solution in that range and (b)$\frac{\partial f_p(x)}{\partial p} < 0$for all interesting$p,x$i.e.$f$is strictly decreasing with$p$for the same$x$– kennytm Mar 20 '17 at 17:43 • The minimal$p$which a solution exists in$x\in(0,\frac\pi2)$is$p=\sqrt5\approx2.23607$, which could be found by solving$f^{(4)}_p(0)=-\frac{4}{p^3}(p^4-6p^2+5)=0$-> could you please explain this in more details? – LCH Mar 20 '17 at 18:58 • @LCH When you decrease$p$, the non-trivial solution will move towards 0. But there already exists a trivial solution at$x=0$with multiplicity 4 (i.e.$f_p(0) = f_p'(0) = f_p''(0) = f_p^{(3)}(0) = 0$). When we reach the minimal$p$, the non-trivial and trivial solutions will "combine" and have multiplicity 5, i.e.$f_p^{(4)}(0) = 0\$. – kennytm Mar 20 '17 at 19:08