Let us define $$f_1(t) = a_1 \cos(w_1 t + \phi_1)$$ and $$f_2(t) = a_2 \cos(w_2 t + \phi_2).$$
The ratio of the periode length is rational, $\frac{\omega_1}{\omega_2} \in \mathbb{Q}$.
Finally, I want to find the maximum point $\left(t^*, f(t^*)\right)$, where $f(t) = f_1(t)+f_2(t)$. I guess $t^*$ must be the function of $a_i$, $w_i$, and $\phi_i$ for $i\in\{1,2\}$, and want to observe the relation between $t^*$ and them.
I've already tried to solve the problem $$\max_t f(t)$$ using some formulas about trigonometric functions and differentiation, but I failed to find $t^*$. Actually, I can find a value using some software (e.g. MATLAB) but I want to find $t^*$ into a closed form equation with respect to $a_i$'s, $w_i$'s and $\phi_i$'s.
Thanks to tell me some materials with respect to this issue or direction to do for the next step.