Identify the function which represents a peri | vedclass

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(D) function $f(t)$ is periodic if it repeats its value after a fixed interval of time $T$,such that $f(t) = f(t + T)$.$1$. The functions $e^{-\omega

Vedclass · Accepted Answer

$\sin \omega t + \cos \omega t$

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(D) function $f(t)$ is periodic if it repeats its value after a fixed interval of time $T$,such that $f(t) = f(t + T)$.$1$. The functions $e^{-\omega t}$,$e^{\omega t}$,and $\log_{e}(\omega t)$ are non-periodic functions because they do not repeat their values at regular intervals.$2$. For the function $f(t) = \sin \omega t + \cos \omega t$,we can check for periodicity by replacing $t$ with $t + T$:$f(t + T) = \sin(\omega(t + T)) + \cos(\omega(t + T)) = \sin(\omega t + \omega T) + \cos(\omega t + \omega T)$.$3$. If we choose $T = \frac{2\pi}{\omega}$,then $\omega T = 2\pi$. Substituting this:$f(t + T) = \sin(\omega t + 2\pi) + \cos(\omega t + 2\pi) = \sin \omega t + \cos \omega t = f(t)$.Since the function repeats itself after a time period $T = \frac{2\pi}{\omega}$,it represents a periodic motion.

Identify the function which represents a periodic motion.

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