Question

Find the period of the following function:

$\sin \frac{x}{2}-\cos \frac{x}{2}$

• A. $\pi$
• B. $2\pi$
• C. $3\pi$
• D. $4\pi$

Answer 1

The correct option is D $4\pi$

$\sin \frac{x}{2}-\cos \frac{x}{2}$

$=\sqrt{2}\left[\frac{1}{\sqrt{2}}\sin \frac{x}{2}-\frac{1}{\sqrt{2}}\cos \frac{x}{2}\right]$

$=\sqrt{2}\left[\cos \frac{\pi }{4}\sin \frac{x}{2}-\sin \frac{\pi }{4}\cos \frac{x}{2}\right]$

$=\sqrt{2}\sin \left(\frac{x}{2}-\frac{\pi }{4}\right)$

$oncomparingitwith$

$=A\sin \left(Bx-c\right)+D$

$wegetB=\frac{1}{2}$

$Period=\frac{2\pi }{\left(\frac{1}{2}\right)}=4\pi$