Question

HELPPP!!

Answer 1

$\bf \textit{Sum and Difference Identities} \\\\ sin(\alpha + \beta)=sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta) \\\\ sin(\alpha - \beta)=sin(\alpha)cos(\beta)- cos(\alpha)sin(\beta) \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf c_2sin(\omega t)+c_1cos(\omega t)~~ \begin{cases} c_2=Acos(\theta )\\ c_1=Asin(\theta ) \end{cases} \\\\\\ Acos(\theta )sin(\omega t)+Asin(\theta )cos(\omega t)\implies Asin(\omega t)cos(\theta )+Acos(\omega t)sin(\theta ) \\\\\\ \stackrel{\textit{common factor}}{A[sin(\omega t)cos(\theta )+cos(\omega t)sin(\theta )]}\implies A[sin(\omega t + \theta) ]\implies Asin(\omega t + \theta)$