# What is the period of f(t)=sin( t / 36 )+ cos( (t)/ 42) ?

##### 2 Answers
Oct 3, 2016

$T = 504 \pi$

#### Explanation:

First of all we, know that $\sin \left(x\right)$ and $\cos \left(x\right)$ have a period of $2 \pi$.

From this, we can deduct that $\sin \left(\frac{x}{k}\right)$ has a period of $k \cdot 2 \pi$: you can think that $\frac{x}{k}$ is a variable running at $\frac{1}{k}$ the speed of $x$. So, for example, $\frac{x}{2}$ runs at half the speed of $x$, and it will need $4 \pi$ to have a period, instead of $2 \pi$.

In your case, $\sin \left(\frac{t}{36}\right)$ will have a period of $72 \pi$, and $\cos \left(\frac{t}{42}\right)$ will have a period of $84 \pi$.

Your global function is the sum of two periodic functions. By definition, $f \left(x\right)$ is periodic with period $T$ if $T$ is the smallest number such that

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

and in your case, this translates into

$\sin \left(\frac{t}{36} + T\right) + \cos \left(\frac{t}{42} + T\right) = \sin \left(\frac{t}{36}\right) + \cos \left(\frac{t}{42}\right)$

From here, you can see that the period of $f \left(x\right)$ can't be $72 \pi$ nor $84 \pi$, because only one of the two terms will make a whole turn, while the other will assume a different value. And since we need both terms to do a whole turn, we need to take the least common multiple between the two periods:

$\lcm \left(72 \pi , 84 \pi\right) = 504 \pi$

Oct 3, 2016

$1512 \pi$.

#### Explanation:

The least positive P (if any ) such that f( t + P ) = f( t ) is befittingly

called the period of f(t). For this P, f(t+nP)=f(t), n =+-1,, +-2, +-3, ....

For $\sin t \mathmr{and} \cos t , P = 2 \pi .$

For $\sin k t \mathmr{and} \cos k t , P = \frac{2}{k} \pi .$

Here,

the period for $\sin \left(\frac{t}{36}\right)$ is pi/18 and,

for $\cos \left(\frac{t}{42}\right)$, it is $\frac{\pi}{21}$.

For the given compounded oscillation f(t), the period P should be

such that it is also the period for the separate terms.

This P is given by P=M(pi/18)=N(pi/21). For M= 42 and N= 36,

$P = 1512 \pi$

Now, see how it works.

$f \left(t + 1512 \pi\right)$

$= \sin \left(\frac{t}{36} + 42 \pi\right) + \cos \left(\frac{t}{42} + 36 \pi\right)$

$= \sin \left(\frac{t}{36}\right) + \cos \left(\frac{t}{42}\right)$

=f(t).

If halve P to 761 and this is odd. So, P = 1512 is the least possible

even multiple of $\pi$.