# What is the period of the trigonometric function given by f(x)=2sin(5x)?

Feb 16, 2015

The period is: $T = \frac{2}{5} \pi$.

The period of a periodic function is given by the period of the function divided the number the multiplies the $x$ variable.

$y = f \left(k x\right) \Rightarrow {T}_{f u n} = {T}_{f} / k$

So, for example:

$y = \sin 3 x \Rightarrow {T}_{f u n} = {T}_{\sin} / 3 = \frac{2 \pi}{3}$

$y = \cos \left(\frac{x}{4}\right) \Rightarrow {T}_{f u n} = {T}_{\cos} / \left(\frac{1}{4}\right) = \frac{2 \pi}{\frac{1}{4}} = 8 \pi$

$y = \tan 5 x \Rightarrow {T}_{f u n} = {T}_{\tan} / 5 = \frac{\pi}{5}$.

In our case:

${T}_{f u n} = {T}_{\sin} / 5 = \frac{2 \pi}{5}$.

The $2$ changes only the amplitude, that, from $\left[- 1 , 1\right]$, becomes $\left[- 5 , 5\right]$.