Question

find the exact value.
\\( \tan ( - 6 \pi ) \\)

Explanation:

Step1: Use the property of tangent function

The tangent function \(y = \tan(x)\) has a period of \(\pi\), i.e., \(\tan(x + n\pi)=\tan(x)\) for any integer \(n\). Also, \(\tan(-x)=-\tan(x)\).

First, \(\tan(-6\pi)=-\tan(6\pi)\). Since \(6\pi = 6\times\pi\) and using the period - property \(\tan(x + n\pi)=\tan(x)\) with \(x = 0\) and \(n = 6\), we know that \(\tan(6\pi)=\tan(0 + 6\pi)\).

Step2: Evaluate \(\tan(0)\)

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

Since \(\tan(-6\pi)=-\tan(6\pi)\) and \(\tan(6\pi)=\tan(0)\) (because of the period - property of the tangent function), then \(\tan(-6\pi)=- 0\).

Answer:

\(0\)