<s>[INST] Switch from $a\cdot \sin(t) + b \cdot \cos(t)$ to $c \cdot \cos(t+f)$ How could I switch from $a\cdot \sin(t) + b \cdot \cos(t)$ to $c \cdot \cos(t+f)$?
Thank you for your time.
 [/INST] $$
a\sin t+b\cos t=\sqrt{a^2+b^2}\left[\frac{b}{\sqrt{a^2+b^2}}\cos t +\frac{a}{\sqrt{a^2+b^2}}\sin t\right]$$
$$
=\sqrt{a^2+b^2}\left[\cos \alpha \cos t +\sin\alpha \sin t\right]$$
$$ =\sqrt{a^2+b^2} \cos(t-\alpha). $$
$ (c,f) $ have different symbols.
Useful in combining two waves of same frequency but different amplitude.
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