Question

Find the exact value. Do not use a calculator.

$\tan \left(6\mathrm{\pi }\right)$

Short answer

The exact value of$\tan \left(6\mathrm{\pi }\right)=0$

Step-by-step solution

 Step 1. Given information

$\tan \left(6\mathrm{\pi }\right)$

Step 2. Unit circle

$$\begin{gathered} \mathrm{Rotate}\mathrm{anticlockwise}\mathrm{to}\mathrm{form}6\mathrm{\pi }\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{positive}\mathrm{x}-\mathrm{axis}. \\ \mathrm{The}\tan \left(6\mathrm{\pi }\right)\mathrm{equals}\mathrm{the}\frac{\mathrm{y}-\mathrm{coordinate}}{\mathrm{x}-\mathrm{coordinate}}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{intersection}(1,0)\mathrm{of}\mathrm{unit}\mathrm{circle}. \end{gathered}$$

Step 3. Final answer

$$\begin{gathered} \tan \left(6\mathrm{\pi }\right)=\frac{\mathrm{y}}{\mathrm{x}}=\frac{0}{1}=0 \\ \tan \left(6\mathrm{\pi }\right)=0 \end{gathered}$$