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How should I begin to add together these two trigonometric functions: $$3\cos\left(3t + \frac\pi5\right) + 4\cos\left(4t+\frac\pi8\right)$$ in order that I might obtain the value of the period of their sum?

I have considered expanding the two cosines, according to the rule of sum, but have gotten very little from it; and have wondered whether one could write the two using complex exponentials: $$3e^{i(3t+π/5)} + 4e^{i(4t+π/8)}.$$

Blue
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V.E.
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  • Instead of expanding the cosines, it might be good to consider the graphs of the two functions and see where their periods overlap. – Chen Jul 17 '21 at 03:37
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    The fundamental period is $2 \pi$ (try replacing $t$ with $t + 2 \pi$). Even $\cos 2x + \sin x$ also has period $2 \pi$. – Toby Mak Jul 17 '21 at 03:52
  • @MichaelHardy you are a very patient individual. – A rural reader Jul 17 '21 at 03:54
  • If the two frequencies were the same, one could write the sum as a single sine wave by applying some standard trigonometric identities. But the frequencies differ, so this is as simple as it gets. – Michael Hardy Jul 17 '21 at 03:54

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\begin{align} & 3\cos(3t + \pi/5) + 4\cos(4t+\pi/8) \\[8pt] = {} & 3\cos(3t)\cos(\pi/5) - 3\sin(3t)\sin(\pi/5) \\ & {} + 4\cos(4t)\cos(\pi/8) - 4\sin(4t)\sin(\pi/8) \\[8pt] = {} & A\cos(3t) + B\sin(3t) + C\cos(4t) + D\sin(4t) \tag 1 \\[8pt] & \text{where } A = 3\cos(\pi/5),\, B=-3\sin(\pi/5), \\ & C= 4\cos(\pi/8), \, D = -4\sin(\pi/8). \end{align}

For some purposes it may be better to write it in the form seen on line $(1)$ above.

But the terms involving $3t$ and those involving $4t$ cannot be combined to make a single sine wave. To see that, look at the graph. It's not shaped like a single sine wave.

Michael Hardy
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