Suppose I have the following function: $$f_{\omega, a, b, c}(t) = 2\omega^2(a\sin(\omega t)^2 + b\sin(2\omega t)^2 + c\sin(3\omega t)^2 - 1)(2a\sin(\omega t)^2 - a - 32 b\sin(\omega t)^4 + 32 b\sin(\omega t)^2 - 4b + 18c\sin(3\omega t)^2 - 9 c) + a\sin(\omega t)^2 + b\sin(2\omega t)^2 + c\sin(3\omega t)^2 - k - (a\sin(\omega t)^2 + b\sin(2 \omega t)^2 + c\sin(3 \omega t)^2)^2,$$ where $k$ is a fixed number.
I want to find the parameters $\omega, a, b, c$ that depend on $k$ s.t. $f_{\omega, a, b, c}$ is approximately zero for all values of $t$ in $[0, 5 \pi]$. I tried to evaluate the function at points $t=0, t = \frac{\pi}{\omega}, t = \frac{\pi}{2\omega}, t = \frac{\pi}{3\omega}$ and equate to zero but got polynomial equations of order 5, so I couldn't find explicit solutions. Could you please help?