Q 30. - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value. Do not use a calculator.

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

Step-by-Step Solution

Answer

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

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

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