Question

The period of $3\sin (2x+\frac{\pi }{3})-4\cos (\pi x-\frac{\pi }{6})$ is

• A. $1$
• B. $\pi$
• C. $2+\pi$
• D. Not defined

Answer 1

The correct option is D Not defined

We know that, period of both $a+b\sin (mx+n)$ and $a+b\cos (mx+n)$ is $\frac{2\pi }{|m|}.$
Hence, period of $3\sin (2x+\frac{\pi }{3})$ is $\pi$ and period of $4\cos (\pi x-\frac{\pi }{6})$ is $2.$

Therefore, period of $3\sin (2x+\frac{\pi }{3})-4\cos (\pi x-\frac{\pi }{6})$
$=\text{LCM of}(\pi ,2)$
$2$ is a rational number and $\pi$ is an irrational number.As the L.C.M of a rational and irrational is not defined.

Hence, period of $3\sin (2x+\frac{\pi }{3})-4\cos (\pi x-\frac{\pi }{6})$ is undefined.