Larry Riddle, Agnes Scott College

### Zeros for Numerator in $$f_\theta (r)$$

Let $$N(\theta) = \sin((m-1)\theta) + \sin(m \theta) -\sin((k-1)\theta) -\sin(k \theta)$$ where $$k = \left\lceil \frac{90^\circ}{\theta}\right\rceil$$ and $$m = \left\lceil \frac{450^\circ}{\theta} \right\rceil$$. We claim that on the interval 0° < θ ≤ 180°, N(θ) = 0 for $$\theta = \frac{540^\circ}{p}$$ for all integers p ≥ 3 and at $$\theta = \frac{360^\circ}{2p+1}$$ for all integers p ≥ 1.

We will make use of two trig identities: $\sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\ \\ \sin(A) - \sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$

Using these identities gives \begin{align} N(\theta) &= \left(\sin((m-1)\theta) + \sin(m \theta)\right) -\left(\sin((k-1)\theta) +\sin(k \theta)\right) \\ \\ &= 2\sin\left(\frac{2m-1}{2}\theta\right)\cos\left(\frac{1}{2}\theta\right) - 2\sin\left(\frac{2k-1}{2}\theta\right)\cos\left(\frac{1}{2}\theta\right) \\ \\ &= 2\cos\left(\frac{1}{2}\theta\right) \left[\sin\left(\frac{2m-1}{2}\theta\right) - \sin\left(\frac{2k-1}{2}\theta\right) \right] \\ \\ &= 4\cos\left(\frac{1}{2}\theta\right)\cos\left(\frac{m+k-1}{2}\theta\right)\sin\left(\frac{m-k}{2}\theta\right). \end{align}

We first consider $$\theta = \dfrac{540^\circ}{p}$$ for integers p in the form p = 6n + b, for n ≥ 0 and b = 0, 1, 2, 3, 4, and 5. (Note that if n = 0, then only b = 3, 4, and 5 give θ ≤ 180°.)

If $$\theta = \dfrac{540^\circ}{6n}$$ for n ≥ 1, then \begin{align} k &= \left\lceil{\frac{90^\circ}{\theta}}\right\rceil = \left\lceil{\frac{90(6n)}{540}}\right\rceil = \lceil{n}\rceil = n \\ \\ m &= \left\lceil\frac{450^\circ}{\theta}\right\rceil = \left\lceil\frac{450(6n)}{540}\right\rceil = \left\lceil{5n}\right\rceil = 5n \end{align}

Hence $\sin\left(\frac{m-k}{2}\theta\right) = \sin\left(\frac{4n}{2}\frac{540^\circ}{6n}\right) = \sin(180^\circ) = 0 \Rightarrow N(\theta) = 0.$ If $$\theta = \dfrac{540^\circ}{6n+b}$$ for n ≥ 0, where b = 1, 2, 3, 4, or 5, then \begin{align} k &= \left\lceil{\frac{90^\circ}{\theta}}\right\rceil = \left\lceil{\frac{90(6n+b)}{540}}\right\rceil = \left\lceil{{n+\frac{b}{6}}}\right\rceil = n+1 \\ \\ m &= \left\lceil\frac{450^\circ}{\theta}\right\rceil = \left\lceil\frac{450(6n+b)}{540}\right\rceil = \left\lceil{5n+\frac{5b}{6}}\right\rceil = 5n+b \end{align} Hence $\cos\left(\frac{m+k-1}{2}\theta\right) = \cos\left(\frac{6n+b}{2}\frac{540^\circ}{6n+b}\right) = \cos(270^\circ) = 0 \Rightarrow N(\theta) = 0.$

Now suppose $$\theta = \dfrac{360^\circ}{2p+1}$$. First consider if p = 2n is even. Then \begin{align} k &= \left\lceil{\frac{90^\circ}{\theta}}\right\rceil = \left\lceil{\frac{90(4n+1)}{360}}\right\rceil = \left\lceil{{n+\frac{1}{4}}}\right\rceil = n+1 \\ \\ m &= \left\lceil\frac{450^\circ}{\theta}\right\rceil = \left\lceil\frac{450(4n+1)}{360}\right\rceil = \left\lceil{5n+\frac{5}{4}}\right\rceil = 5n+2 \end{align} Hence $\sin\left(\frac{m-k}{2}\theta\right) = \sin\left(\frac{4n+1}{2}\frac{360^\circ}{4n+1}\right) = \sin(180^\circ) = 0 \Rightarrow N(\theta) = 0.$ Now if p = 2n+1 is odd, then \begin{align} k &= \left\lceil{\frac{90^\circ}{\theta}}\right\rceil = \left\lceil{\frac{90(4n+3)}{360}}\right\rceil = \left\lceil{{n+\frac{3}{4}}}\right\rceil = n+1 \\ \\ m &= \left\lceil\frac{450^\circ}{\theta}\right\rceil = \left\lceil\frac{450(4n+3)}{360}\right\rceil = \left\lceil{5n+\frac{15}{4}}\right\rceil = 5n+4 \end{align} Hence $\sin\left(\frac{m-k}{2}\theta\right) = \sin\left(\frac{4n+3}{2}\frac{360^\circ}{4n+3}\right) = \sin(180^\circ) = 0 \Rightarrow N(\theta) = 0.$

Here is a graph of the zeros for 40° ≤ θ ≤ 180°. Notice that between each pair of red zeros are three black zeros.