How Is Trigonometry Related To Polynomials?

$\begin{aligned} f(x) & = x^8 - x^6 + x^4 - x^2 + 1 \quad \quad \small \color{#3D99F6}{\text{Let }z = x^2} \\ & = z^4 - z^3 + z^2 - z + 1 \\ & = \frac{z^5+1}{z+1} \end{aligned}$

For $f(x) = 0$ , we have:

$\begin{aligned} \frac{z^5+1}{z+1} & = 0 \\ \Rightarrow z^5 & = -1 \quad \small \text{except for }z = -1 \text{ or } x = \pm i \text{, when }f(x) \text{ is undefined.} \\ x^{10} & = e^{i(2k+1)\pi} \\ \Rightarrow x & = e^{i\frac{(2k+1)\pi}{10}} \quad \small \text{where } k = 0, 1, \color{#D61F06}{\not{2}}, 3, 4,5,6,\color{#D61F06}{\not{7}},8,9; \color{#D61F06}{\text{when } k = 2, 7 \space \Rightarrow x = \pm i} \\ & = \cos \left(\frac{2k+1}{10}\pi \right) + i \sin \left(\frac{2k+1}{10}\pi \right) \end{aligned}$

The largest imaginary part $q = \sin \frac{3\pi}{10} = \sin \boxed{54}^\circ$

Since x=0 is not a solution, divide the equation by x^4,

then make it a quadratic in x^2+1/x^2