Question

Find the exact value. Do not use a calculator.

$\tan (-3\mathrm{\pi })$

Short answer

The exact value of$\tan (-3\mathrm{\pi })=0$

Step-by-step solution

Step 1. Given information

$\tan (-3\mathrm{\pi })$

Step 2. Unit Circle

$$\begin{gathered} \mathrm{Rotate}\mathrm{anticlockwise}\mathrm{to}\mathrm{form}-3\mathrm{\pi }\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{positive}\mathrm{x}-\mathrm{axis}. \\ \mathrm{The}\tan (-3\mathrm{\pi })\mathrm{equals}\mathrm{the}\frac{\mathrm{y}-\mathrm{coordinate}}{\mathrm{x}-\mathrm{coordinate}}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{intersection}(-1,0)\mathrm{of}\mathrm{unit}\mathrm{circle}. \end{gathered}$$

Step 3. Final Result

$$\begin{gathered} \tan (-3\mathrm{\pi })=-\frac{0}{-1}=0 \\ \tan (-3\mathrm{\pi })=0 \end{gathered}$$