Q. 72 - Precalculus Enhanced with Graphing Utilities

Question

Use a calculator to find the approximate value of each expression rounded to two decimal places: $\csc \frac{5\pi }{13}$

Step-by-Step Solution

Answer

The approximate value of the expression $\csc \frac{5\pi }{13}$ is $1.07$.

Step 1. Given information

We are given an expression $\csc \frac{5\pi }{13}$

We need to find the approximate value of the expression that is rounded to two decimal places.

Step 2. Concept

• Note that $\csc \theta = \frac{1}{\sin \theta }$
• While finding the values of the trigonometric functions using a calculator, we need to set the correct MODE in the calculator depending on whether the angle is in radian or in degree.

Step 3. Finding the approximate value of the given expression

The given trigonometric expression is, $\csc \frac{5\pi }{13}$

Since the angle is in radian, we have to set the correct MODE in the calculator to receive radian.

Most of the calculators do not have $\csc$ key. Therefore we will make use of the relation, $\csc \theta = \frac{1}{\sin \theta }$

Therefore, 

$$\begin{gathered} \csc \left(\frac{5\pi }{13}\right)=\frac{1}{\sin \left(\frac{5\pi }{13}\right)} \\ \Rightarrow \csc \left(\frac{5\pi }{13}\right)\approx 1.07 \end{gathered}$$

which is rounded off two two decimal places.

Step 4. Final Answer

The approximate value of the expression $\csc \left(\frac{5\pi }{13}\right)=1.07$