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     This part answered :-

How does one determine what is $t$ when we have two trigonometric functions added to each other? Like $\tan(x) + \sin(2x)$ \begin{align} \tan(x) &= \tan(x + \pi) \\ \sin(2x) &= \sin(2x + \pi) \end{align}

Do we take the $\pi$ as $t$ because they have it in common?

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Is the sum of trigonometric periodic function often periodic or always and why?(i would like to share my take on this one from what i read and Deduced)

And how do we determine that this function is periodic and find $t$? $$ \tan(x) \sin(3x) $$ since \begin{align} \tan(x) &= \tan(x + \pi) \\ \sin(3x) & = \sin(2x + 2\pi/3) \end{align} And thx in advance

lodo
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    The period of the sum of 2 functions is the least common multiple of their period – Math Attack Sep 20 '23 at 20:25
  • It probably makes some sense to change the title to reflect you are interested in this question specifically over finite sums of periodic functions. – Snared Sep 20 '23 at 20:30
  • Will do , @Snared – lodo Sep 20 '23 at 20:32
  • @MathAttack Mind sharing some examples to make sure I understood correctly – lodo Sep 20 '23 at 20:36
  • @lodo There are numerous answers of varying technical detail posted on this related question: https://math.stackexchange.com/questions/681750/sum-of-two-periodic-functions-is-periodic – Snared Sep 20 '23 at 20:42

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I read the resources above and tried to understand the two participants shares , I understand now how to determine t for the sum of two functions , As said above we try to find the least common multiple of their period . for example:-

$\sin(2x) + \cos(3x)$ \begin{align} \cos(3x) &= \cos(3x +2\pi/3)&=\cos(3x +tn)\\ \sin(2x) &= \sin(2x + \pi) &= \sin(2x +tm)\\ \end{align} $m>0 ,n>0$ and they are whole numbers

So in this case we have the least common multiple when n is 3 and m is 2 Resulting in t equals $2\pi $ for the sum of the two functions .

I will try to share what I have deduced in regards to if the sum of two Periodic functions is always periodic since I now have a slightly better understanding of the topic ,

The answer is no, there does not exist non zero integers $a$,$b$ such that $a t_1=bt_2$

For example cos($\pi x$)+sin($2x$). We can see that $a \times 2=b \times \pi$ does not exist, Hence the answer is a non periodic function

If what I deduced is incorrect any correction is appreciated,

I am left with a couple of questions unanswered, I will try to edit the original version to highlight them.

Dheeraj Gujrathi
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lodo
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