# calculating the amplitude of a cosine function

I want to be able to be able to get the amplitude of the following function:

$$||A||\cos(2 \omega t + a)+||B||\cos(3 \omega t +b)+||C||\cos(5 \omega t +c)$$

I am trying to find a way to get the amplitude of this function. This is usually simple for when there is one value of $\omega$ to consider, but I'm having a hard time thinking how to go about this.

One thing I did try was to just sum $||A||, ||B||, ||C||$ together. If we ignore the phases a, b, c, this gives an idea of what the max value obtained is, but I don't think it aptly describes the amplitude of the function as it is not sinusoidal in the traditional sense and most of the time, is below this max value. I noted that that min value is not equal to the negative of the max value either.

I played around with the RMS $$\sqrt{||A||^2+||B||^2+||C||^2}$$ but i'm sure if that is an appropriate approach.

Suggestions?

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If you are thinking to solve this as $A\cos({k\omega t+\phi})$ and you are asking for what's the value of $A$ as real number. Its not possible. Amplitude is time varying function in this case and it is hard to simplify all three terms into one term. If you had two terms, You can see how amplitude varies with time. –  Ramana Venkata May 10 '12 at 20:13
Where does $D$ come from? –  Daan Michiels May 10 '12 at 21:29
Why the notation $\| A \|$ etc. in the first displayed formula? –  Francesco Sica May 10 '12 at 21:38
The essential problem is that there is no well-defined notion of "amplitude" for a function that is not a pure sinusoid. You can consider the RMS value of the function instead, $\big(\frac1{T}\int_0^T f(x)^2 dx\big)^{1/2}$. –  Rahul May 10 '12 at 22:02
Are you the same user as Suzuki who asked the very similar math.stackexchange.com/questions/139505/… –  Ross Millikan May 11 '12 at 0:17