Question

1. consider the function $f(x) = \sin x$.\

a. is this a periodic function? if so, what is its period?

Explanation:

Step1: Recall the definition of a periodic function

A function \(y = f(x)\) is periodic if there exists a positive number \(T\) such that \(f(x + T)=f(x)\) for all \(x\) in the domain of \(f\).
For the function \(y = \sin x\), we know that \(\sin(x + 2k\pi)=\sin x\), where \(k\in\mathbb{Z}\).

Step2: Find the fundamental period

The smallest positive value of \(T\) for which \(\sin(x + T)=\sin x\) holds for all \(x\).
Using the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\), if \(\sin(x + T)=\sin x\), then \(\sin x\cos T+\cos x\sin T=\sin x\) for all \(x\).
This implies \(\cos T = 1\) and \(\sin T=0\). The smallest positive \(T\) that satisfies these equations is \(T = 2\pi\).

Answer:

Yes, \(y = \sin x\) is a periodic function and its period is \(2\pi\).