What is the period of f(t)=sin( t /6 )+ cos( (7t)/24 ) ?

May 12, 2018

The period is $= 12 \pi$

Explanation:

A periodic function $f \left(x\right)$ is such that

$f \left(x\right) = f \left(x + T\right)$

where, $T$ is the period

Here,

$f \left(t\right) = \sin \left(\frac{t}{6}\right) + \cos \left(\frac{7}{24} t\right)$

$f \left(t + T\right) = \sin \left(\frac{1}{6} \left(t + T\right)\right) + \cos \left(\frac{7}{24} \left(t + T\right)\right)$

$= \sin \left(\frac{1}{6} t + \frac{1}{6} T\right) + \cos \left(\frac{7}{24} t + \frac{7}{24} T\right)$

Therefore,

$f \left(t\right) = f \left(t + T\right)$

$\left\{\begin{matrix}\sin \left(\frac{t}{6}\right) = \sin \left(\frac{1}{6} t + \frac{1}{6} T\right) \\ \cos \left(\frac{7}{24} t\right) = \cos \left(\frac{7}{24} t + \frac{7}{24} T\right)\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\sin \left(\frac{t}{6}\right) = \sin \left(\frac{t}{6}\right) \cos \left(\frac{1}{6} T\right) + \cos \left(\frac{t}{6}\right) \sin \left(\frac{1}{6} T\right) \\ \cos \left(\frac{7}{24} t\right) = \cos \left(\frac{7}{24} t\right) \cos \left(\frac{7}{24} T\right) - \sin \left(\frac{7}{24} t\right) \sin \left(\frac{7}{24} T\right)\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\cos \left(\frac{1}{6} T\right) = 1 \\ \sin \left(\frac{1}{6} T\right) = 0 \\ \cos \left(\frac{7}{24} T\right) = 1 \\ \sin \left(\frac{7}{24} T\right) = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{1}{6} T = 2 \pi \\ \frac{7}{24} T = 2 \pi\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}T = 12 \pi \\ T = \frac{48}{7} \pi = 48 \pi\end{matrix}\right.$

The LCM of $12 \pi$ and $48 \pi$ is $= 12 \pi$

The period is $= 12 \pi$

graph{sin(x/6)+cos(7x/24) [-8.3, 56.63, -15.6, 16.88]}

May 13, 2018

$48 \pi$

Explanation:

Period of $\sin \left(\frac{t}{6}\right) - \to 6 \left(2 \pi\right) = 12 \pi$
Period of $\cos \left(\frac{7 t}{24}\right) - \to \frac{24 \left(2 \pi\right)}{7} = \frac{48 \pi}{7}$
Period of f(t) is the least common multiple of $12 \pi \mathmr{and} \frac{48 \pi}{7}$.
$12 \pi$ .....x (4) ........ --> $48 \pi$
$\frac{48 \pi}{7}$ ...x (7) .... --> $48 \pi$

Period of f(t) --> $48 \pi$