Exact solution to the the equation $(2 \pi - \theta)\cos \theta + \sin \theta = 0$

Question

I have been trying to solve the following goat grazing problem:

A goat is tied to the outside of a circular fence. If the length of the rope is the same as the circumference of the fence, what is the maximum area upon which the goat can feed?

I'm using integration to calculate the area but to find the limits for the integral, I need to solve the following transcendental equation: $$(2 \pi - \theta)\cos \theta + \sin \theta = 0$$

Is it possible to compute the exact solution of this equation or do I just have to solve it numerically?

Answer 1

To solve the equation for $\theta=1.7897758\dots$, use $2\pi-\theta=x$:

$$(2\pi-\theta)\cos(\theta)+\sin(\theta)=0\iff \tan(x)=x$$

Now use the series solution in

Rearrange $y = x\tan y$ to solve for $y$ given $x$,

Bessel J zero $j_{v,x}$, and Stirling S1 $S_n^{(m)}$ to get:

$$\bbox[5px,border: 5px groove black]{\theta=\frac\pi2-\sum_{m=1}^\infty\sum_{n=0}^m\frac{S_m^{(n)}\Gamma(n+m-1)(-1)^ni^{m+n}2^{m-n}}{\left(\frac{3\pi}2-i\right)^{n+m-1} m!\Gamma(m)}=\pi-\tan^{-1}(j_{\frac32,1})}$$

shown here is the series and the closed form also works

Answer 2

If, as @Tyma Gaidash did, you let $2\pi-\theta=x$, you nedd to find the zero's of function $$f(x)=x-\tan(x)$$ whic is better to write as $$g(x)=x\cos(x)-\sin(x)$$ in order to remove the discontinuities.

Because of the multiplying $x$, the solutions are closer and closer to $(2n+1)\frac \pi 2$.

Expanding as series around this value an simplifying the trigonometric functions since $n$ is an integer. This gives $$g(x)=\sum_{k=0}^\infty \frac{(-1)^n \left(2 (k-1) \cos \left(\frac{\pi k}{2}\right)-\pi (2 n+1) \sin \left(\frac{\pi k}{2}\right)\right)}{2 \Gamma (k+1)}\, \left(x- (2 n+1)\frac{\pi}{2} \right)^k$$

Truncate to some order and use power series reversion to obtain $$x_{(n)}=q-\frac{1}{q}-\frac{2}{3 q^3}-\frac{13}{15 q^5}-\frac{146}{105q^7}-\frac{781}{315q^9}+O\left(\frac{1}{q^{11}}\right)$$ where $q=(2 n+1)\frac{\pi}{2}$.

For the first root, the above truncated series would give $x_{(1)}=\color{red}{4.493409}66$ while the "exact" solution is $4.49340946$.