Trig Identity question. To Determine A and B in the identity: $e ^{( A + Bi ) t} = (\cos5t + i \sin5t )e ^{− t }$

Question

Just wondering how to solve this basic identity. I have attempted separating out the e value on one side with the trig function on the opposite side of the equals sign. I am now unable to find a trig identity to simplify out to find out what $A$ and $B$ is. Any help would be greatly appreciated :)

Answer 1

Hint: Use that $$\cos(5t)+i\sin(5t)=e^{5it}$$

Answer 2

Hint: Since $$(\cos 5t + i \sin 5t) = e^{5it}$$ $$e^{5it}*e^{-t} = e^{t(5i-1)} = e^{t(-1+5i)}$$

Can you compare this result to $e^{A+Bi}$?