Q 26. - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value. Do not use a calculator.

$\sec \left(8\mathrm{\pi }\right)$

Step-by-Step Solution

Answer

The exact value of$\sec \left(8\mathrm{\pi }\right)=1$

Step 1. Given information

$\sec \left(8\mathrm{\pi }\right)$

Step 2. Unit Circle:

$$\begin{gathered} \mathrm{Rotate} \mathrm{anticlockwise} \mathrm{to} \mathrm{form} 8\mathrm{\pi } \mathrm{angle} \mathrm{with} \mathrm{the} \mathrm{positive} \mathrm{x}-\mathrm{axis}. \\ \mathrm{The} \sec \left(8\mathrm{\pi }\right) \mathrm{equals} \mathrm{the} \frac{1}{\mathrm{x}-\mathrm{coordinate}\left(1\right)} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} (1, 0) \mathrm{of} \mathrm{unit} \mathrm{circle}. \end{gathered}$$

Step 3. Final Result

$$\begin{gathered} \sec \left(8\mathrm{\pi }\right)=\frac{1}{\mathrm{x}-\mathrm{coordinate}}=\frac{1}{1}=1 \\ \sec \left(8\mathrm{\pi }\right)=1 \end{gathered}$$