Finding period of rational function of trigonometric functions

Question

I want to find the period of the function $$f(x)=\frac{2\cos (3x)\cdot |\sin(5x)|\cdot 3|\tan (8x)|\cos^4(x)}{7\tan\left( \frac{x}{9}\right)\cdot \sec^3\left(\frac{2x}{3}\right)}$$ which is a rational function of trig functions.

So far I have found that:

• the period of $\displaystyle \cos(3x)$ is $\frac{2\pi}{3};$

• the period of $\displaystyle |\sin (5x)|$ is $\frac{\pi}{5};$

• the period of $\displaystyle \cos^4(x)$ is $\pi$;

• the period of $\displaystyle \tan(x/9)$ is $9\pi$;

• and the period of $\displaystyle \sec^3(2x/3)$ is $3\pi$.

How do I find the total period?

Answer 1

Each of the functions are $2 \pi k$ periodic, each for different $k$ (since $f(x)$ is a rational function of trig functions). You are missing a few, so I will fill in the full list:

• For $\cos(3x)$, $k = 1/3$,
• For $|\sin(5x)|$, $k = 1/10$,
• For $|\tan(8x)|$, $k = 1/16$,
• For $\cos^4(x)$, $k = 1/2$,
• For $\tan(x/9)$, $k = 9/2$,
• For $\sec^3(2x/3)$, $k = 3/2$.

Then the question is what is the least common multiple of all of these numbers (Somos above said $\gcd$, I think you want $\mathrm{lcm}$). Why the $\mathrm{lcm}$? The least common multiple tells you what is the smallest number that is divisible by each of the above $k$. That is because each of the functions will are periodic with the least common multiple. We want the smallest since if $f(x)$ is $T$ periodic, it is also $2T, 3T, \ldots$ periodic. But we want the period $T$.

For the $k$ values listed above, the $\mathrm{lcm}$ is $9$. So the function is $18\pi$ periodic. You can check this using your favorite software by checking $| f(x) - f(x+18\pi) |$ as long as you avoid the $x$-values for which the function blows up.