Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers how to determine the inclination of a line from its slope by using the arctan function, how to find the angle between two nonparallel lines with the tangent difference formula, and how to compute the distance from a point to a line using a specific formula. The material reinforces the relationship between slope and trigonometric functions and emphasizes real-world applications, such as designing inclined transportation systems and measuring road grades.

Learning Objectives

1

Determine the inclination of a nonhorizontal line using its slope and the arctan function.

2

Compute the angle between two intersecting lines using their slopes.

3

Apply the distance formula to find the perpendicular distance from a point to a line.

4

Connect the geometric concepts of slope, inclination, and angle measurement to real-world applications such as road grades and inclined railways.

Key Concepts

CONCEPT

DEFINITION

Inclination of a Line

The positive angle (less than π/2 radians) measured counterclockwise from the positive x-axis to the nonhorizontal line.

Slope

A measure of the steepness of a line, defined as the rate of change of y with respect to x. For a nonvertical line with inclination θ, the slope m is given by m = tan(θ).

Angle Between Two Lines

The smallest positive angle formed at the intersection of two nonparallel lines, typically found by comparing the lines’ inclinations or using the formula tan(φ) = |(m2 - m1)| / (1 + m1*m2).

Distance from a Point to a Line

The length of the perpendicular segment from a given point to a line, computed using the formula d = |Ax₁ + By₁ + C| / √(A² + B²) for a line in the form Ax + By + C = 0.

Example Problems

Example 1

The ________ of a nonhorizontal line is the positive angle $\theta$ (less than $\pi$) measured counterclockwise from the $x$-axis to the line.

Example 2

If a nonvertical line has inclination $\theta$ and slope $m$, then $m =$ ______________ .

Example 3

If two nonperpendicular lines have slopes $m_1$ and $m_2$ the angle between the two lines is $\tan\ \theta =$ ___________ .

Example 4

The distance between the point $(x_1, y_1)$ and the line $Ax +By + C = 0$ is given by $d =$ ________ .

Example 5

In Exercises 5-12, find the slope of the line with inclination $\theta$.
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Step-by-Step Explanations

QUESTION

Find the inclination of the line given by y = mx + b, where m is known.

STEP-BY-STEP ANSWER:

Step 1: Identify the slope (m) from the equation of the line.
Step 2: Use the relation m = tan(θ) to set up the equation tan(θ) = m.
Step 3: Solve for the inclination (θ) by taking the arctan of m, i.e., θ = arctan(m).
Final Answer: The inclination of the line is θ = arctan(m) (in radians).

Finding the Inclination of a Line

QUESTION

Given two lines with slopes m₁ and m₂, how do you find the angle (φ) between them?

STEP-BY-STEP ANSWER:

Step 1: Use the formula tan(φ) = |(m₂ - m₁)| / (1 + m₁*m₂).
Step 2: Compute the numerator as the absolute difference of the slopes.
Step 3: Compute the denominator as 1 plus the product of the slopes.
Step 4: Find φ by calculating φ = arctan(|(m₂ - m₁)| / (1 + m₁*m₂)).
Final Answer: The angle between the two lines is φ = arctan(|(m₂ - m₁)| / (1 + m₁*m₂)).

Finding the Angle Between Two Lines

QUESTION

How do you find the perpendicular distance from the point (x₁, y₁) to the line with equation Ax + By + C = 0?

STEP-BY-STEP ANSWER:

Step 1: Identify the coefficients A, B, and C from the line equation.
Step 2: Substitute the coordinates (x₁, y₁) into the formula d = |Ax₁ + By₁ + C| / √(A² + B²).
Step 3: Compute the absolute value of the numerator and find the square root of the sum of the squares of A and B.
Final Answer: The perpendicular distance is d = |Ax₁ + By₁ + C| / √(A² + B²).

Finding the Distance from a Point to a Line

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Common Mistakes

  • Confusing the direction of measurement by using the y-axis instead of the x-axis for the angle of inclination.
  • Forgetting to take the absolute value when calculating the angle between two lines.
  • Neglecting to convert the arctan result from radians to degrees when needed for practical applications.
  • Misinterpreting a negative slope as having a negative inclination, instead of understanding that inclination is always a positive measure (for nonhorizontal lines).
  • Mixing up the concept of the slope with the rate of change without connecting it to the tangent function.