i got a few questions...
how is (e^3x)sin2x=(e^3x)Im(e^2ix)?
im not very good in complex numbers...
and then
to find d56/dt56 [(e^-t)(sint)]
im confused by
(-1+i)^56=[ √2 e^(3 πi/4)]^56
=(2^28)cos42 π + isin42 π
→(-1+i)^56=2^28 <<<<how do you get this??
For part 2, are you asking how you go from
to ?
If so, it's done using DeMoivre's Theorem.
DeMoivre's Theorem states that:
If
Then for .
So to evaluate , convert to polars and use DeMoivre's Theorem.
.
We can see that and .
The function is in the second quadrant.
So
.
Therefore
Now use DeMoivre's Theorem:
.
Now notice that the angle is the same as the angle , since you have just gone around the unit circle 21 times.
So
.