Question

Finding the Period Give a convincing argument that the period of tan \(x\) is \(\pi .\)

Short answer

The tan function, tan\(x\), repeats itself every \(\pi\) radians, therefore its period is \(\pi\).

Step-by-step solution

Definition of Periodic Function

A periodic function is a function that repeats its values in regular intervals. So, a function \(f(x)\) is said to be periodic if there exists a non-zero real number \(t\) such that \(f(x + t) = f(x)\) for all \(x\). All trigonometric functions are periodic.

Consider Function tan\(x\)

The tan function is a trigonometric function which is defined as the ratio of the sine to the cosine of an angle \(x\), i.e., tan \(x = \frac{sin(x)}{cos(x)}\). It is undefined wherever cos\(x\) = 0, namely at odd multiples of \(\frac{\pi}{2}\). So, it goes from \( - \infty \) to \( \infty \) as \(x\) moves from \(\frac{-\pi}{2}\) to \(\frac{\pi}{2}\). Beyond this, the function repeats itself every \(\pi\) radians.

Periodicity of tan\(x\)

Let's consider the behaviour of the tan function for two specific values of the argument: tan\(x\) and tan\(x+\pi\). Since tan\(x = \frac{sin(x)}{cos(x)}\), we get tan\(x+\pi = \frac{sin(x+\pi)}{cos(x+\pi)}\). Using the properties of sine and cosine functions, sin\((x + \pi) = sin(x)\) and cos\((x + \pi) = cos(x)\). Therefore, tan\((x + \pi) = \frac{sin(x)}{cos(x)}\), which is tan\(x\). Hence, the period of tan\(x\) is \(\pi\).