Complex Amplitudes

The questions below are due on Thursday September 18, 2025; 02:00:00 PM.
 
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Part a. Find a complex constant c_1 so that Re(c_1e^{j\omega t})=\sin(\omega t) for all real numbers \omega and t.

c_1 =~

Part b. Find a complex constant c_2 so that Re(c_2e^{j\omega t})=\cos(\omega t)+\sin(\omega t) for all real numbers \omega and t.

c_2 = ~

Part c. Find a complex constant c_3 so that Re(c_3e^{j\omega t})=A\cos(\omega t)+B\sin(\omega t) for all real numbers \omega and t.

c_3 = ~

Part d. Find a complex constant c_4 so that Re(c_4e^{j\omega t})=\cos(\omega t−\phi) for all real numbers \omega, t, and \phi.

c_4=~

Part e. Find a complex constant c_5 so that Re(c_5e^{j\Omega n})=\cos((2\pi−\Omega)n) for all real numbers \Omega and all integers n.

c_5 =~