Chapter 5.1, Problem 103E

(a)

To determine

To demonstrate: The given identity for $\theta ={132}^{\text{o}}$ , using the table feature of a graphing utility.

Explanation of Solution

Given:

The identity:

  ${\csc }^2\theta -{\cot }^2\theta =1$ , when $\theta ={132}^{\text{o}}$

Graph:

As per the given problem

Consider the left hand side of the equation

Put $\theta ={132}^{\text{o}}$ in the given identity

  ${\csc }^2{132}^{\text{o}}-{\cot }^2{132}^{\text{o}}$

Graph the two terms individually.

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{y}}_1={\csc }^2{132}^{\text{o}}$ and ${\text{y}}_2={\cot }^2{132}^{\text{o}}$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

The table feature of the graphing utility shows the two expressions.

Subtract ${\text{y}}_2$ from ${\text{y}}_1$

  ${\text{y}}_1-{\text{y}}_2=1.810727-0.810727=1$

Interpretation: According to the above table, it can be observed that ${\text{y}}_1-{\text{y}}_2=1$ .

Therefore, the identity ${\csc }^2\theta -{\cot }^2\theta =1$ is verified for $\theta ={132}^{\text{o}}$ .

(b)

To determine

To demonstrate: The given identity for $\theta =\frac{2\pi }{7}$ , using the table feature of a graphing utility.

Explanation of Solution

Given:

The identity:

  ${\csc }^2\theta -{\cot }^2\theta =1$ , when $\theta =\frac{2\pi }{7}$

Graph:

As per the given problem

Consider the left hand side of the equation

Put $\theta =\frac{2\pi }{7}$ in the given identity

  ${\csc }^2\frac{2\pi }{7}-{\cot }^2\frac{2\pi }{7}$

Graph the two terms individually.

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{y}}_1={\csc }^2\frac{2\pi }{7}$ and ${\text{y}}_2={\cot }^2\frac{2\pi }{7}$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

The table feature of the graphing utility shows the two expressions.

Subtract ${\text{y}}_2$ from ${\text{y}}_1$

  ${\text{y}}_1-{\text{y}}_2=1.635964-0.635964=1$

Interpretation: According to the above table, it can be observed that ${\text{y}}_1-{\text{y}}_2=1$ .

Therefore, the identity ${\csc }^2\theta -{\cot }^2\theta =1$ is verified for $\theta =\frac{2\pi }{7}$ .