# What is the period of f(theta) = tan ( ( 13 theta)/4 )- cos ( ( 6 theta)/ 5 ) ?

Apr 18, 2018

$\frac{20 \pi}{39}$

#### Explanation:

$\tan x$ has a period of $\pi$, while $\cos x$ has a period of $2 \pi$

Thus $\tan \left(\frac{13 \theta}{4}\right)$ completes one period, when $\theta$ changes by $\frac{4 \pi}{13}$, while

$\cos \left(\frac{6 \pi}{5}\right)$ repeats when $\theta$ changes by $\frac{5 \pi}{3}$

So, the period of $f \left(\theta\right)$ is the least common multiple of $\frac{4 \pi}{13}$ and $\frac{5 \pi}{3}$, i.e. $\frac{20 \pi}{39}$