# Plotting complex sinusoid to a cosine wave

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ssmith on 15 Jan 2023
Commented: Star Strider on 16 Jan 2023
I have the complex exponential equation (z) and I want to adjust the plot to produce a cosine wave of amplitude of 5.0 that is shifted by 45 degrees. I have attached my code so far below. If someone could direct me to where I should make adjustments to plot a cosine wave with a = 5.0 shifted by 45 degrees that would be greatly appreciated. Thank you.
z = exp(-j*2*pi*t);
clear all; close all;
Tt = 1; % Total time
fs = 500; % Sampling frequncy
t = (0:1/fs:Tt); % Time vector
a = 5.0; % Amplitude
z = exp(-j*2*pi*t); % Complex sinusoid
plot(t,real(z),'k',t,imag(z),':k'); % Plot result
xlabel('Time (sec)', 'FontSize',14);
ylabel('y(t)','FontSize',14);

Star Strider on 15 Jan 2023
Add to the complex argument to shift the complex exponential result by 45°
Tt = 1; % Total time
fs = 500; % Sampling frequncy
t = (0:1/fs:Tt); % Time vector
a = 5.0; % Amplitude
z = exp(-1j*(2*pi*t+pi/4)); % Complex sinusoid
plot(t,real(z),'k',t,imag(z),':k'); % Plot result
xlabel('Time (sec)', 'FontSize',14);
ylabel('y(t)','FontSize',14);
.
ssmith on 16 Jan 2023
Oh ok, so to follow the exact phrase, your method is correct, but my 'z' also works?
Star Strider on 16 Jan 2023
Not quite, since it multiplies both the real and imaginary parts of ‘z’ by ‘a’.
To be absolutely rigorous, the real and imaginary parts of ‘a’ need to be multiplied respectively by the real and imaginary parts of ‘z’.
Your Question specified that the cosine (real) part of the complex exponential function was to be multiplied by ‘a’. It just depends on how much detail (and how rigorously) you want to define this.
I leave that to your discretion.

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