Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 5.2, Problem 89E

To determine

To explain: Why the given equation is not an identity and find one value of the variable for which the equation is not true.

Expert Solution & Answer

Explanation of Solution

Given information:

The following equation:

$sec2x−1=tanx$

Formula used:

The following identity:

$1+tan2x=sec2x$

The equation can be simplified as:

$sec2x−1=tanxtan2x=tanx [Pythagorean identity]±tanx≠tanx [Square root]$

The left hand side is not equal to the right hand side. Hence the equation is not an identity.

Let $x=2π3$

Substituting value of $x$ on left hand side of the equation,

$sec2x−1=sec22π3−1 [Substitute value of x]=(−2)2−1 [Put value of sec 2π3]=4−1 [Square]=3 [Subtract]$

Substituting value of $x$ on right hand side of the equation,

$tanx=tan2π3 [Substitute value of x]=−3 [Put value of tan2π3]$

The value on the left hand side is not equal to the value on the right hand side.

Hence for $x=2π3$ , the equation is not true

Also, the given equation is not an identity because the expressions on the left hand side and the right hand side when plotted on a graph, do not coincide.

Graph:

Sketch the graph using graphing utility.

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

Step 2: Press $Y=$ button.

Step 3: Enter the expressions $Y1=sec2x−1$ and $Y2=tanx$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

Precalculus with Limits: A Graphing Approach, Chapter 5.2, Problem 89E

Interpretation:

The graphs of the left hand side expression and the right hand side expression do no coincide.

Chapter 5.2, Problem 88E

Chapter 5.2, Problem 90E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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