In summary, to find the imaginary part of the function C = A*e^(-i*wt)*sin(k*x), use Euler's formula eiθ = cos(θ) + i sin(θ) to expand the expression and then simplify it. The coefficient of i will be the imaginary part of C. After substituting the expanded expression back into the original, use the distributive law of multiplication to expand the brackets. The resulting expression will be of the form C = R + iI, where I is the imaginary part of C.
Yes, substitue it back into the original expression, and then expand out the brackets using the distributive law of multiplication, i.e. A(B + C) = AB + AC.
Then you will have an expression of the form C = R + iI, and I is the imaginary part of C.