# Period of sums/multiplied sines/cosines

• Jul 10th 2011, 02:10 AM
gabber808
Period of sums/multiplied sines/cosines
Hello

I am trying to revise some rules regarding the period of sines (or cosines) when (a) they are added , and (b) they are multiplied. I haven't had too much luck Googling this topic.

For example, what would be the fundamental period of:

y=cos(2.pi.2x) + cos(2.pi.3x)

Or, if we multiply:

y=cos(2.pi.2x).cos(2.pi.3x)

People have suggested to me that I use trig. identities (or complex exponentials) but I am sure there is a more intuitive way such as considering the phases between both parts.

Any ideas appreciated !

Cheers

Gabber
• Jul 10th 2011, 07:37 AM
kalyanram
Re: Period of sums/multiplied sines/cosines
Quote:

Originally Posted by gabber808
People have suggested to me that I use trig. identities (or complex exponentials) but I am sure there is a more intuitive way such as considering the phases between both parts.

Consider $f(x) = cos(2\pi .2x) + cos(2\pi .3x)$ if $f(x)$ is periodic then $f(x+T) = f(x)$ where $T$ is a period.

Now we have $cos(2\pi .2(x+T)) + cos(2\pi .3(x+T)) = cos(2\pi .2x) + cos(2\pi .3x)$ as $cosine$ is a function with periodicity $2\pi$ we must have $2\pi(2T)$ and $2\pi(3T)$ are both of the form $2m\pi$ where $m$ is an integer not necessarily same in both the cases. In first case its enough if $T = \frac{m}{2}$ ,and in the second case if $T = \frac{n}{3}$ where $m,n \in Z$ where $Z$ is the set of integers. Therefore if $T=1$ to be the fundamental period then we are fine with it.

Multiplication of functions can be handled on the same lines.

Now try solving the following problem instead
$f(x) = cos(2\pi .\frac{x}{2}) + cos(2\pi .\frac{x}{3})$

Kalyan.
• Jul 10th 2011, 11:05 AM
gabber808
Re: Period of sums/multiplied sines/cosines
Thank you Kaylan

So it looks like the period will be the LCM (lowest common multiple) of the individual periods?

Do you have any good ideas how to find the period when the cosines are multiplied?

Thanks again

Gabber
• Jul 10th 2011, 11:25 AM
kalyanram
Re: Period of sums/multiplied sines/cosines
Quote:

So it looks like the period will be the LCM (lowest common multiple) of the individual periods?

That's true period in the case of sums is the LCM of individual periods.

Quote:

Do you have any good ideas how to find the period when the cosines are multiplied?
Now in the case when cosines are multiplied or for that matter any powers of sines or cosines can be handled by transforming them to sums of sines and cosines.

Kalyan.
• Jul 10th 2011, 11:53 AM
kalyanram
Re: Period of sums/multiplied sines/cosines
Here are few examples
$f(x) = 2cos(2\pi . \frac{x}{2}).cos(2\pi . \frac{x}{3}) \Rightarrow f(x) = cos(2\pi . \frac{5x}{6}) + cos(2\pi . \frac{x}{6})$ as you can see since the LCM has already been obtained during the sum and difference of angles the approach does not change drastically. However in the following example the approach varies

$f(x) = cos^3 (2\pi . \frac{x}{12}) \Rightarrow f(x) = \frac{1}{4} (3cos(2\pi x) - cos(2\pi . \frac{x}{4}))$ , $T = 4$.

Kalyan
• Jul 10th 2011, 12:30 PM
mike999
Re: Period of sums/multiplied sines/cosines
--------------------------
For example, what would be the fundamental period of:
y=cos(2.pi.2x) + cos(2.pi.3x)
Or, if we multiply:
y=cos(2.pi.2x).cos(2.pi.3x)
-----------------------------
my anwer is:
in both cases the total period is
x=(0;1)
that is the least common multiple of the two periods.
in fact the 1st period is x=(0;1/2)
and the 2nd period is x=(0;1/3)
therefore the range x=(0;1) is the least common multiple of the two single ranges.
bye-bye
mike999
• Jul 10th 2011, 10:59 PM
gabber808
Re: Period of sums/multiplied sines/cosines
Thanks everybody

Mike, you seem to be saying that this LCM method is also true for procucts of sines (or cosines), not just when they are added. So is this generally true?

Kalyan, in your cos-cubed example, after you transform it, we have periods of 1 and 4. So again it is the LCM.

Thanks again

Matt
• Jul 11th 2011, 03:43 AM
kalyanram
Re: Period of sums/multiplied sines/cosines
Hi Matt,

Quote:

Mike, you seem to be saying that this LCM method is also true for procucts of sines (or cosines), not just when they are added. So is this generally true?

Kalyan, in your cos-cubed example, after you transform it, we have periods of 1 and 4. So again it is the LCM.
The idea of taking an LCM is valid for the case of sum or multiplication of functions. All I was suggestting is that you have to consider taking the transformations when multiplication of functions is involved as in the case of $cos^2 \theta, cos^3 \theta$ are involved.

Kalyan.
• Jul 11th 2011, 11:53 AM
gabber808
Re: Period of sums/multiplied sines/cosines
Hello again

Thank you for all your help so far Kalyan

However, I have just been trying an online graphing program.

Firstly I tried cos(2pi*(x/2))*cos(2pi*(x/3)). As expected I can see that the pattern repeats every 6 units.

Then I tried cos(2pi*x)*cos(2pi*(x/3)). Maybe I am wrong, but it looks like a period of 1.5. There is ALSO a period of 3. But the lowest common multiple (LCM) idea does not suggest that !

Does this mean that perhaps the LCM idea is not the way to go and really we should always use trig identies to get the period when our sines or cosines are multiplied?
• Jul 11th 2011, 12:04 PM
kalyanram
Re: Period of sums/multiplied sines/cosines
Hi Gbber808,
Quote:

Then I tried cos(2pi*x)*cos(2pi*(x/3)). Maybe I am wrong, but it looks like a period of 1.5. There is ALSO a period of 3. But the lowest common multiple (LCM) idea does not suggest that !
Well all our previous examples we dealt with simple cases as of the type $\frac{x}{m} , \frac{x}{n}$ where we took the $lcm(m,n)$ in the case $f(x) = cos(2\pi x).cos(2 \pi \frac{x}{3}) = \frac{1}{2}(cos(2\pi \frac{4x}{3}) + cos(2 \pi \frac{2x}{3}))$ and we see that its enough for $T =n \frac{3}{2}$ for $cos(2\pi \frac{4x}{3}) , cos(2\pi \frac{2x}{3})$ to be periodic.

Kalyan.
• Jul 11th 2011, 12:12 PM
kalyanram
Re: Period of sums/multiplied sines/cosines
Another way to look at it is that you have taken the $lcm(1,3)$ i.e LCM of denominators and divided it by HCF of numerators $hcf(2,4)$
T = 3/2.

Kalyan.
• Jul 13th 2011, 09:53 AM
gabber808
Re: Period of sums/multiplied sines/cosines
Sorry for the late reply. Apart from lots of work I have also been revising my trig identities. It was a long ago I learnt those. Thanks again for all of your help Kalyanram.
• Apr 21st 2015, 12:30 PM
Re: Period of sums/multiplied sines/cosines
Hi
How
f(x) = cos^3 (2pi*x/12) = 1/4 (3cos(2pi* x) - cos(2pi*x/4)) ??

As According to the trigonometric identity (cos^3(X) = (3/4)cosX + (1/4)cos(3X) ) it should be

1/4 (3cos(2pi*x/12) - cos(6pi*x/12))

I am missing some point here ,
kindly explain

Thanks
• Apr 21st 2015, 03:24 PM
kalyanram
Re: Period of sums/multiplied sines/cosines
Quote:

Originally Posted by kalyanram
$f(x) = cos^3 (2\pi . \frac{x}{12}) \Rightarrow f(x) = \frac{1}{4} (3cos(2\pi x) - cos(2\pi . \frac{x}{4}))$ , $T = 4$

There was a mistake in the formula as pointed out by Mask.
$f(x) = cos^3 (2\pi . \frac{x}{12}) \Rightarrow f(x) = \frac{1}{4} (3cos(2\pi \frac{x}{12}) - cos(2\pi . \frac{x}{4}))$ , and hence $T = 12$

~K