Existence of roots of $A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$

Question

It seems very intuitive that $$f(t)=A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$$ has roots, but how to prove it?

$A_i>0$, $\omega_i>0$ and $\phi_i\geqslant0$ (even though these restrictions are not necessary).

• I'm not able to find $t_1$ and $t_2$ so that $f(t_1)$ and $f(t_2)$ have opposite signs (intermediate value theorem);
• I tried using complex exponentials but end up with the same expression appearing in the argument;
• I wrote $\omega_1t+\phi_1=-\arcsin(\dots)$ but it does not help (and the converse requires attention);
• I considered $f'$ but don't see any perspective.

It could be easier to prove if $\dfrac{\omega_1}{\omega_2}\in \mathbb{Q}$, but I'm looking for a general proof.

Any clues?

Answer 1

Hint: Suppose that $A_1=\max\{A_1,A_2\}$, then take $t_0$, such that $\omega_1t_0+\phi_1=\frac{\pi}{2}$ and take $t_1$, such that $\omega_1t_1+\phi_1=\frac{3\pi}{2}$, then:

$$f(t_0)>0$$ and
$$f(t_1)<0$$ How $f$ is continuous, then there is a root in $[t_0,t_1]$