# What is the frequency of f(theta)= sin 6 t - cos 2 t ?

Jun 6, 2016

It is $\frac{1}{\pi}$.

#### Explanation:

We look for the period that is easier, then we know that the frequency is the inverse of the period.

We know that the period of both $\sin \left(x\right)$ and $\cos \left(x\right)$ is $2 \pi$. It means that the functions repeat the values after this period.

Then we can say that $\sin \left(6 t\right)$ has the period $\frac{\pi}{3}$ because after $\frac{\pi}{3}$ the variable in the $\sin$ has the value $2 \pi$ and then the function repeats itself.

With the same idea we find that $\cos \left(2 t\right)$ has period $\pi$.

The difference of the two repeats when both quantities repeat.
After $\frac{\pi}{3}$ the $\sin$ start to repeat, but not the $\cos$. After $2 \frac{\pi}{3}$ we are in the second cycle of the $\sin$ but we do not repeat yet the $\cos$. When finally we arrive to $\frac{3}{\pi} / 3 = \pi$ both $\sin$ and $\cos$ are repeating.

So the function has period $\pi$ and frequency $\frac{1}{\pi}$.

graph{sin(6x)-cos(2x) [-0.582, 4.283, -1.951, 0.478]}