# Do we always take the imaginary part when solving for sine and the real part when solving for cosine?

I’m currently working on differential equations with particular solutions by guessing a complex solution that includes both a real and an imaginary part. For example, for the following Problem, I have written out a new complex differential equation and guessed a complex solution (See work below):

Original equation: y ‘’(x)+2y’(x)+y(x)=sin(x).

New complex differential equation: Y’’(x) + 2Y ‘(x)+Y(x)=Y(0)e^(ix), where Y(0) is a constant

Taking the derivatives and solving for Y(0), I got that Y(0) = 1/2i. The characteristic equation was -Y(0)+2iY(0)+Y(0)=1.

At the very end, I have that my complex differential equation, written out with sines and cosines, is: Y(x) = (-1/2)icosx +(1/2)sinx.

Originally, I thought that the answer would be obtained by taking the Real part, but the answer in the book is the Imaginary Part ((-1/2)cosx). Is it always the case that you use the Imaginary Part for a sine function and the Real part for a cosine function?

Also, when do you guess a solution that is a complex solution vs. not - is guessing a complex solution useful only with trig functions/waves?

Thanks very much.