Chapter 5.1, Problem 104E

(a)

To determine

To demonstrate: The given identity for $\theta ={346}^{\text{o}}$ , using the table feature of a graphing utility.

Explanation of Solution

Given:

The identity:

  ${\tan }^2\theta +1={\sec }^2\theta$ , when $\theta ={346}^{\text{o}}$

Graph:

As per the given problem

Put $\theta ={346}^{\text{o}}$ in the given identity

  ${\tan }^2{346}^{\text{o}}+1={\sec }^2{346}^{\text{o}}$

Graph the two sides of the equation as individual expressions.

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 ${\text{y}}_1={\tan }^2{346}^{\text{o}}+1$ and ${\text{y}}_2={\sec }^2{346}^{\text{o}}$ , 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.

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

Therefore, the identity ${\tan }^2\theta +1={\sec }^2\theta$ is verified for $\theta ={346}^{\text{o}}$ .

(b)

To determine

To demonstrate: The given identity for $\theta =3.1$ , using the table feature of a graphing utility.

Explanation of Solution

Given:

The identity:

  ${\tan }^2\theta +1={\sec }^2\theta$ , when $\theta =3.1$

Graph:

As per the given problem

Put $\theta =3.1$ in the given identity

  ${\tan }^2(3.1)+1={\sec }^2(3.1)$

Graph the two sides of the equation as individual expressions.

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 ${\text{y}}_1={\tan }^2(3.1)+1$ and ${\text{y}}_2={\sec }^2(3.1)$ , 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.

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

Therefore, the identity ${\tan }^2\theta +1={\sec }^2\theta$ , is verified for $\theta =3.1$ .