Question

The period of $\left|\sin \frac{x}{2}\right|+\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|$ is

• A. $\frac{\pi }{2}$
• B. $2\pi$
• C. $\pi$
• D. $4\pi$

Answer 1

The correct option is D $4\pi$

We know that the period of $|\sin x|$ and $|\cos x|$ is $\pi .$

The period of
$a+b\sin (mx+n)$ and $a+b\cos (mx+n)$ is $\frac{2\pi }{|m|}$

Using the above information, we get the period of
$\left|\sin \frac{x}{2}\right|\to 2\pi$
$\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|\to 4\pi$

Now, the period of
$\left|\sin \frac{x}{2}\right|+\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|$
$=\text{LCM of}(2\pi ,4\pi )$
$=4\pi$