Question

express 2sin(3t) + 2sqrt(3)cos(3t) as Asin(Bt+C) where A > 0

Answer 1

If you have an expression of the form: \[a \sin \theta + b \cos \theta\]
and you want to rewrite as: \[A \sin (\theta +\alpha)\] Then you need to know two formulas: \[A = \sqrt{a ^{2}+b ^{2}}\] \[\alpha = \tan^{-1} \frac{ b }{ a }\] Thus, your expression can be rewritten as:
\[4\sin (3t+\frac{ \pi }{ 3 })\]Lastly, to double-check your work you can recall the angle addition formula for the sine function: \[\sin (\alpha+\beta) = \sin (\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\] and then see if your result matches the equation you started with. Hope this helps!