# What is the period of f(theta) = tan ( ( 5theta)/12 )- cos ( ( 2 theta)/ 3 ) ?

Jul 23, 2016

$12 \pi$

#### Explanation:

The period of $\tan k \theta$ is $\frac{\pi}{k}$

and the period of $\cos k \theta$ is $\frac{2 \pi}{k}$.

So, here,

the separate periods of the two terms in $f \left(\theta\right)$ are

$\frac{12 \pi}{5} \mathmr{and} 3 \pi$.

For $f \left(\theta\right)$, the period P is such that $f \left(\theta + P\right) = f \left(\theta\right)$,

both the terms are become periodic and P is the least possible such

value.

Easily, $P = 5 \left(\frac{12}{5} \pi\right) = 4 \left(3 \pi\right) = 12 \pi$

Note that, for verification,

$f \left(\theta + \frac{P}{2}\right) = f \left(\theta + 6 \pi\right)$ is not $f \left(\theta\right)$, whereas

$f \left(\theta + n P\right) = f \left(\theta + 12 n \pi\right) = f \left(\theta\right) , n = 1 , 2 , 3 , . .$