# What is the period of f(t)=sin( ( 5 t) /4 )?

Dec 21, 2015

$f \left(t\right) = \sin \left(\frac{5 t}{4}\right)$ has a period of $\frac{8 \pi}{5}$

#### Explanation:

$\sin \left(\theta\right)$ has a period (i.e. a pattern that repeats every increment) of $2 \pi$

For $\sin \left(\frac{\theta}{2}\right)$, $\theta$ would need double the incremental distance to reach the repetition point.
i.e. $\sin \left(\frac{\theta}{2}\right)$ would have a period of $2 \times 2 \pi$

and
$\sin \left(\frac{\theta}{4}\right)$ would have a period of $4 \times 2 \pi = 8 \pi$

Similarly we can see that
$\sin \left(5 \cdot \theta\right)$ would have a period of $\frac{2 \pi}{5}$

Combining these two observations (and replacing $\theta$ with $t$)
we have
$\textcolor{w h i t e}{\text{XXX}} \sin \left(\frac{5 t}{4}\right)$ has a period of $2 \pi \cdot \frac{4}{5} = \frac{8 \pi}{5}$