Trigonometric Periodicity

Geometry Level 2

$\large \color{#69047E}{g(x)=\cos|x|+\sin|x|}$

Find the fundamental period of the function $\color{#69047E}{g(x)}$ .

\dfrac{\pi}{2}

\pi

2\pi

4\pi

g(x)

is not periodic None of the above

1 solution

A Former Brilliant Member
May 28, 2015

So, for finding the fundamental period of $g\left( x \right)=\cos { \left| x \right| } +\sin { \left| x \right| }$ , we need to find the LCM of time periods of both $\sin { \left| x \right| }$ and $\cos { \left| x \right| }$

So, clearly from the graph of $\cos { \left| x \right| }$ :

We can see that it has a fundamental period of $2\pi$ .

But, if we see the graph of
$\sin { \left| x \right| }$ :

It can be seen that $\sin { \left| x \right| }$ has no fundamental period and therefore it is non-periodic.

Now, as $\cos { \left| x \right| }$ is periodic but, $\sin { \left| x \right| }$ isn't, so we can infer that $g\left( x \right)$ is also non-periodic.

Moderator note:

Is it a necessity to solve this problem by graphing?

It's actually not necessary, noticing that $\cos |x|$ is equivalent to $\cos(x)$ while it can be easily shown that $\sin |x|$ is not periodic. Sum of these two can't be periodic. Do you see why?

To the challenge master....Well it is easy for someone like you, me and many others to see tell in one look which is a periodic function and which is not. But, for someone who is a novice, it is difficult, and so graphing helps quite a bit. Isn't that the entire purpose of a solution??....No Offence

A Former Brilliant Member - 6 years ago

I got confused between $\sin |x| + \cos |x|$ and $|\sin x| + |\cos x|$ :(

Shubhrajit Sadhukhan - 3 months ago

Trigonometric Periodicity

1 solution

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