Sum of two sinusoids in order to write it on new standard form

Question

Is it correct that if I am given

and I have to find $A_2$, $ω_2$ and $φ_2$ in order to write it on the standard form below

that I have to add $f_1(t)$ and $f_2(t)$ together, and then set that equal to the standard form above?

Would the following below be the correct method to use, in order to find the new values $A_2$, $ω_2$ and $φ_2$ where I finish off by using trigonometric identities?

Also, why is the new function, which we have to find the new values for, called $f_2$ and not $f_3$?

EDIT: Does $ω_2$ and $φ_2$ just equal $ω_2 - ω_1$ and $φ_2 - φ_1$?

Answer 1

Yes, this is correct. You should be able to see that $\omega_2=\frac \pi 2$. Now you can expand all the sums on both sides and collect the terms in $\sin (\frac \pi 2t)$ and $\cos (\frac \pi 2t)$ to get two equations in $A_2,\varphi_2$