I'm trying to do some refreshing of differential equations featuring damped systems. Specifically, I have a question regarding the differential equation solution to an under damped system involving complex roots. Referring to the attached pdf, an under damped system will yield a complex conjugate pair of roots. I am curious as to why the basic real solution features a sine term (refer to second attachment). If I remember Euler's formula correctly, the sine term is always imaginary and is not featured in the real solution. However, this document states otherwise... I believe I have a fundamental misconception regarding this topic. Any idea why the negative imaginary conjugate yields a sine term in the time domain?