Question

find the exact value of ( \tan(-2pi) ). do not use a calculator

( \tan(-2pi)=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the property of tangent function

The tangent function has the property $\tan(-\alpha)=-\tan\alpha$. So, $\tan(-2\pi)=-\tan(2\pi)$.

Step2: Use the periodicity of tangent function

The period of the tangent function $y = \tan x$ is $\pi$. That is, $\tan(x + n\pi)=\tan x$ for any integer $n$. For $x = 0$ and $n = 2$, we have $\tan(2\pi)=\tan(0 + 2\pi)=\tan(0)$.

Step3: Find the value of $\tan(0)$

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

Answer:

$0$