Question

find the exact value. tan 5π

Explanation:

Step1: Use the periodicity of the tangent function

The tangent function \(y = \tan x\) has a period of \(\pi\), i.e., \(\tan(x + n\pi)=\tan x\) for any integer \(n\).
For \(x = 0\) and \(n = 5\), we have \(\tan(5\pi)=\tan(0 + 5\pi)\).
Since \(\tan(x + n\pi)=\tan x\), then \(\tan(5\pi)=\tan(0)\).

Step2: Recall the value of \(\tan(0)\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). When \(\theta = 0\), \(\sin(0)=0\) and \(\cos(0) = 1\). So \(\tan(0)=\frac{\sin(0)}{\cos(0)}=\frac{0}{1}=0\).

Answer:

\(0\)