# complex numbers and euler's rule

I have an equation in the time domain

$$A||H(s)||\sin(\omega t+\angle H(s))$$

I understand I can use the Euler equation for the sin term here. However the solution I have says the result after applying the Euler equation is

$$\frac{AH(s) e^{j\omega t}-(AH(s))^{*}e^{-j\omega t} }{2j}$$

I'm not quite sure why there is a complex conjugate of $AH(s)$. I thought the equation would just end up with $AH(s)$ multiplied with the euler formula for sine?

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It is tacitly assumed here that $A$ is real. Let $\angle(H(s))=\theta(s)$; then $H(s)=|H(s)|e^{i\theta(s)}$. By Euler's formula $$\sin\bigl(\omega t+\theta(s)\bigr)={e^{i(\omega t+\theta(s))}-e^{-i(\omega t+\theta(s))}\over 2i}\ ,$$ and therefore $$A\ |H(s)|\sin\bigl(\omega t+\theta(s)\bigr)={A H(s)e^{i\omega t}-(AH(s)^* e^{-i\omega t} \over 2i}\ .$$