# How to calculate the angular frequency of a cosine wave and sine wave added together?

I'm trying to find a way to calculate the angular frequency of the following wave:

$$3\cos(2t)-2\sin(4t-1)$$

I know how to calculate the angular frequency for a cosine wave or sine wave by taking the coefficient of $t$, does it involve trigonometric identities? How do I go about getting the angular frequency of the sum of a cosine and a sine wave as above?

I plotted it on the graph and I could find it that way but would prefer to find it mathematically.

Thanks

• I would like to define angular frequency for a periodic function as $2\pi/T$ where $T$ is the period, but many sources (such as en.wikipedia.org/wiki/Angular_frequency) seem to define angular frequency only for pure sinusoidal functions. What definition are you using? Dec 10, 2022 at 17:41

If the frequencies are slightly different however we can have a $beat$ frequency.
$\begin{array}\\ 3\cos(2t)-2\sin(4t-1) &=3\cos(2t)-2(\sin(4t)\cos(1)-\cos(4t)\sin(1))\\ &=3\cos(2t)-2((2\sin(2t)\cos(2t)\cos(1)-(\cos^2(2t)-\sin^2(2t)\sin(1))\\ \end{array}$