Question

The period of the function $f\left(x\right)=\sin \left(4x\right)+\cos \left(4x\right)$ is:

• A. $\mathrm{\pi }$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $2\mathrm{\pi }$
• D. None of these

Answer 1

The correct option is B

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

Explanation for the correct option:

Compute the period of the given function.

Given function, $f\left(x\right)=\sin \left(4x\right)+\cos \left(4x\right)$

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 $\sin \left(4x\right)$ is $\frac{2\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}$.

Similarly,

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

Then the period of $\cos \left(4x\right)$ is $\frac{2\mathrm{\pi }}{4}=\frac{\mathrm{\pi }}{2}$ .

So, period of $f\left(x\right)=$LCM of periods of $\sin \left(4x\right)$ and $\cos \left(4x\right)$

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

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

Thus, the period of $f\left(x\right)=\sin 4x+\cos 4x$ is $\frac{\mathrm{\pi }}{2}$.

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