Find the period of the function $g(t)=(\cos 2t, \cos 3t)$.

Question

I need a little help with this problem. I says:

Find the period of the function $g:\mathbb{R}\rightarrow\mathbb{R}^{2}$ defined by $g(t)=(\cos 2t, \cos 3t)$.

Ok, so I found that the period of the coordinate functions $f_1(t)=\cos 2t$ is $\pi$ and $f_2(t)=\cos 3t)$ is $\frac{2\pi}{3}$. How do I write the period of the functio g(t) with the period of its coordinate fucntions?

Answer 1

Let's write down a few terms.

Multiples of $\pi$ are $$\pi, 2\pi \,\ldots$$

Multiples of $\frac{2\pi}3$ are $$\frac{2\pi}{3}, \frac{4\pi}3, 2\pi\ldots$$

Hence, the period is $2\pi$.

Answer 2

$\cos(t)$ has period $2\pi$, $\cos(nt)$ has period $\frac{2\pi}{n}$ as you figured out yourself. Now for the period of your $g(t)$ you have to find the least common multiple of $\pi$ and $\frac{2\pi}{3}$. This is $2\pi$.