Real part of a complex exponential with a complex amplitude

Question

I have $z=(x+yi)e^{it}$ and am stuck trying to find the real part.

I tried applying Euler's identity, but I'm pretty sure that's the wrong way to approach this problem. What am I missing?

$$z=(x+yi)\cos{t}+(x+yi)i\sin{t}$$

Answer 1

Assuming $t$ is real, then you are on the right track $$z=(x+yi)\cos{t}+(x+yi)i\sin{t}$$ Multiplying everything $$z=x\cos t+iy\cos{t}+x\sin t-y\sin{t}$$ Then grouping the real term and the imaginary one $$z=(x\cos t -y\sin t) +i(x\sin t +y\cos t)$$