MATLAB: Equation for the sum of sinusoids

sinusoid

Hello,
I am attempting to create a sum of sines waveform using 4 different frequencies, with the same amplitude of 1, and all with a phase of 0. Sampling rate of 60 with a duration of 30 seconds. Through reading different forums I came up with the code below, however I'm not sure I understand the pi/3, pi/4, etc. used in the x1-x4 equations below. Can anyone explain why this is included? or if there is a better more simple formula than what I have below.
Fs = 60;
Ts = 1/60;
t = 0:Ts:30
x1 = 1*cos(2*pi*.73*t+pi/3);
x2 = 1*cos(2*pi*1.33*t-pi/4);
x3 = 1*cos(2*pi*1.93*t+pi/5);
x4 = 1*cos(2*pi*2.93*t-pi/6);
x = x1 + x2 + x3 + x4;
u = numel(x);
M = zeros(u,6);
M(:,3) = x;
M(:,(1:6)~=3);
M
figure(2)
subplot(5,1,1)
h = plot(t,x1); box off; grid off
xlabel('Time(s)');
ylabel('Amplitude');
subplot(5,1,2)
h = plot(t,x2); box off; grid off
xlabel('Time(s)');
ylabel('Amplitude');
subplot(5,1,3)
h = plot(t,x3); box off; grid off
xlabel('Time(s)');
ylabel('Amplitude');
subplot(5,1,4)
h = plot(t,x4); box off; grid off
xlabel('Time(s)');
ylabel('Amplitude');
subplot(5,1,5)
h = plot(t,x); box off; grid off
xlabel('Time(s)');
ylabel('Amplitude');

Best Answer

"I'm not sure I understand the pi/3, pi/4, etc. used in the x1-x4 equations below. Can anyone explain why this is included?"
A quick tutorial
figure; 
t = 0 : 1/60 : 30;
plot(cos(t), 'k-')
hold on
The plot produced above is your base sinusoid. To change the frequency, multiply t by a value greater than one to increase the frequency or a value less than one to decrease the frequency. 
plot(cos(t * 1.2), 'b-')  %increase the freq.
To change the phase, add or subtract values from t. Adding values to t will shift the sinusoid leftward; subtractive values will shift if rightward. Try the block below. 
figure
plot(cos(t), 'k-')
hold on
plot(cos(t + pi/6), 'b:')  %shift leftward
You're working in radians so pi/6 is 30 degrees. If you shift it 360 degrees (2*pi in radians) the black and blue curves will overlap.
In your code (below), t is multiplied by 2*pi*.73 for whatever reason which changes the frequency of your sinusoid. The result is then added to pi/3 which shifts the phase. The reason the values are in fractions of pi is because you're working in radians. 
x1 = 1*cos(2*pi*.73*t+pi/3);
"is a better more simple formula than what I have below"
That entirely depends on what you're doing. If you're just trying to create some sinusoids and their parameters don't matter, you could replace the 2*pi*.73 with whatever value that equals.
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