Q 25. - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value. Do not use a calculator.

$\csc \frac{11\mathrm{\pi }}{2}$

Step-by-Step Solution

Answer

The exact value of$\csc \frac{11\mathrm{\pi }}{2}=-1$

Step 1. Given information

$\csc \frac{11\mathrm{\pi }}{2}$

Step 2. Unit Circle:

$$\begin{gathered} \mathrm{Rotate} \mathrm{anticlockwise} \mathrm{to} \mathrm{form} \frac{11\mathrm{\pi }}{2} \mathrm{angle} \mathrm{with} \mathrm{the} \mathrm{positive} \mathrm{x}-\mathrm{axis}. \\ \mathrm{The} \csc \frac{11\mathrm{\pi }}{2} \mathrm{equals} \mathrm{the} \frac{1}{\mathrm{y}-\mathrm{coordinate}(-1)} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} (0, -1) \mathrm{of} \mathrm{unit} \mathrm{circle}. \end{gathered}$$

Step 3. Final Result

$$\begin{gathered} \csc \frac{11\mathrm{\pi }}{2}=\frac{1}{\mathrm{y}-\mathrm{coordinate}}=\frac{1}{-1}=-1 \\ \csc \frac{11\mathrm{\pi }}{2}=-1 \end{gathered}$$