Question

Which of the following function(s) is/are periodic?

• A. $\sin x+\cos x$
• B. $\cos x+\left\{\frac{x}{\pi }\right\}$
• C. $\cos \pi x+\left\{2x\right\}$
• D. $\ln \left\{x\right\}+\sin 2x$

Answer 1

Answer: (A), (B), (C)

$\cos x+\left\{\frac{x}{\pi }\right\}$
$\sin x+\cos x$
$\cos \pi x+\left\{2x\right\}$

(A) $sinx+cosx=\sqrt{2}sin\left(\frac{\pi }{4}+x\right)\to$ Periodic

(B) cos x $\to$ period $2\pi$

$\left\{\frac{x}{\pi }\right\}\to$ period $\pi$

So period of cos $x+\left\{\frac{x}{\pi }\right\}=2\pi$

(C) $cos\pi x\to$ period 2

$\left\{2x\right\}\to$ period $\frac{1}{2}$

so period of cos $\pi x+\left\{2x\right\}=2$

(D) $ln\left\{x\right\}\to$ period 1

$sin2x\to$ period $\pi$

$ln\left\{x\right\}+sin2x\to$ no period.

since L.C.M of $\pi$ and $1$ does not exist.