Solving Polynomial in $\sin$ and $\cos$ Given the following equation:
$$A \sin(t) + B\cos(t) =C\sin(t)\cos(t).$$
Can we solve the above equation without the need of using general solution of 4-order polynomial equation. We can use computer but not using the search algorithm.
Please help me, this problem can from the minimization of the following function:
$$f(t) = (A + R_1\cos{t})^2 + (B+ R_2\sin{t})^2.$$
that is the problem of finding minimum distance from the origin of coordinate to a ellipse.
in which, $A$, $B$,$C$, $R_1$, $R_2$ is constant. $t$ is variable.
thank you so much.

Let us consider the equation $$A \sin(t) + B\cos(t) -C\sin(t)\cos(t)=0$$ for which we should use tangent half-angle substitution $\big(x=\tan(\frac t2)\big)$. So, after simplifications, we get $$B x^4-2  (A+C)x^3- 2(A- C)x-B=0$$ which is your quartic.
For solving it, I should use Newton method which, starting from a guess $x_0$ will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Strating with $x_0=0$, the first iterate is $$x_1=\frac{B}{2 (C- A)}$$
I do not think that you could obtain an explicit solution for the problem.
