Question

The period of the function $F\left(x\right)=|\sin \left(2x\right)|+|\cos \left(8x\right)|$ is:

• A. $2\mathrm{\pi }$
• B. $\mathrm{\pi }$
• C. $2\frac{\mathrm{\pi }}{3}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Compute the period of the given function.

$F\left(x\right)=|\sin \left(2x\right)|+|\cos \left(8x\right)|$

Let $f\left(x\right)=\left|\sin \left(2x\right)\right|$

We know that the period of $\sin \left(x\right)$ is $2\mathrm{\pi }$.

The general formula for period of $\sin \left(ax\right)$ is $\frac{2\mathrm{\pi }}{a}$.

Then the period of $\sin \left(2x\right)$ is $\frac{2\mathrm{\pi }}{2}=\mathrm{\pi }$ .

Thus the period of $|\sin \left(2x\right)|=\frac{\mathrm{\pi }}{2}$ .

Now, Let $g\left(x\right)=\left|\cos \left(8x\right)\right|$

We know that the period of $\cos \left(x\right)$ is $2\mathrm{\pi }$.

Then the period of $\cos \left(8x\right)$ is $\frac{2\mathrm{\pi }}{8}$.

Thus the period of $\left|\cos \left(8x\right)\right|=\frac{2\mathrm{\pi }}{8\times 2}=\frac{\mathrm{\pi }}{8}$.

$F\left(x\right)=f\left(x\right)+g\left(x\right)=\left|\sin \left(2x\right)\right|+\left|\cos \left(8x\right)\right|$

So, period of $F\left(x\right)=$LCM of periods of $f\left(x\right)$ and $g\left(x\right)$.

Now, we need to calculate the LCM of $\left(\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{8}\right)$ , which is given as:

$\frac{\mathrm{LCM}\mathrm{of}\mathrm{numerators}}{\mathrm{HCF}\mathrm{of}\mathrm{denominators}}$ $=\frac{\pi }{2}$

Thus, the period of $F\left(x\right)=|\sin \left(2x\right)|+|\cos \left(8x\right)|$ is $\frac{\mathrm{\pi }}{2}$.

Hence, Option$\left(\mathrm{D}\right)$ is the correct answer.