Q 23. - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value. Do not use a calculator.

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

Step-by-Step Solution

Answer

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

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

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