Problem 52 - Calculus Single Variable

Question

Let \(y=m x+b\) be the equation of a nonhorizontal, nonvertical line. Such a line intersects the \(x\) -axis making an angle \(\phi\) with the positive \(x\) -axis that has radian measure between 0 and \(\pi .\) What is the relationship between \(m\) and \(\phi ?\) Can this relationship, suitably interpreted, be extended to vertical or horizontal lines? Why is \(\phi\) independent of \(b\) ?

Step-by-Step Solution

Answer

The relationship is \( m = \tan(\phi) \). \( \phi \) is independent of \( b \) because only slope \( m \) affects the angle.

Step 1: Understanding the slope-make relationship

In the equation of the line \( y = mx + b \), the slope \( m \) measures the steepness of the line. When a line intersects the \( x \)-axis, the angle \( \phi \) between the line and the positive \( x \)-axis can be found by relating it to the slope \( m \). Specifically, \( m = \tan(\phi) \), where \( \phi \) is the angle of inclination.

Step 2: Using Trigonometry to find the angle

The slope \( m \) of the line is the tangent of the angle \( \phi \). This gives the relationship \( m = \tan(\phi) \). Therefore, if you have the slope of the line, you can find the angle \( \phi \) using \( \phi = \arctan(m) \). This holds true for all non-vertical and non-horizontal lines.

Step 3: Extending to Vertical and Horizontal Lines

For a horizontal line, \( m = 0 \) and \( \phi = 0 \), which means there is no incline concerning the \( x\)-axis. In the case of a vertical line, \( m \) approaches infinity, which implies that \( \phi = \frac{\pi}{2} \). These special cases fit into the broader framework by appropriately interpreting \( \arctan(m) \) or the limits for extreme angles.

Step 4: Independence from the y-intercept

The angle \( \phi \) is determined solely by the slope \( m \) of the line, which is independent of the \( y \)-intercept \( b \). Changing \( b \) shifts the line up or down, but does not influence its angle of inclination relative to the \( x \)-axis.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in geometry and algebra. It describes how steep a line is. Understanding the slope is crucial as it tells us how a line tilts or inclines. In the equation of a line given by \( y = mx + b \), the slope is represented by \( m \).

Most simply, the slope is the ratio of the vertical change to the horizontal change between any two points on the line. To put it another way:

• If a line rises steeply, \( m \) is high.
• If a line is flat, \( m \) approaches zero.

When a line tilts downward as it moves from left to right, we have a negative slope. On the contrary, a line angled upwards from left to right has a positive slope.

In graphical terms, for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope \( m \) is calculated as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This measure tells us how quickly \( y \) changes with respect to \( x \), and thus, gives the line its direction.

Trigonometric Functions

Trigonometric functions play a pivotal role in understanding angles and their relationships with the sides of triangles. In the context of the angle of inclination of a line, the trigonometric function often used is the tangent function.

Let's break down some key points:

• The tangent of an angle \( \phi \), represented as \( \tan(\phi) \), is the ratio of the opposite side to the adjacent side in a right triangle.
• For a line with slope \( m \), this translates to the equation \( m = \tan(\phi) \).

This connection between slope and trigonometric functions allows us to determine the angle of a line relative to the positive \( x \)-axis. If you know the slope \( m \), you can calculate the angle \( \phi \) by taking the arctangent of \( m \):

\[ \phi = \arctan(m) \]

This reveals the angle's size in radians and is particularly practical for non-vertical, non-horizontal lines. Trigonometric functions thus provide a solid method for understanding how line slopes translate into angular inclinations.

Equation of a Line

The equation of a line is fundamental in coordinate geometry. A common form of a linear equation is \( y = mx + b \). This equation encapsulates the relationship between \( x \) and \( y \) coordinates on a line.

Here's a breakdown of its components:

• \( y \) and \( x \) are the coordinates on the line, representing the vertical and horizontal positions, respectively.
• \( m \) is the slope, which informs us about the line's steepness and direction.
• \( b \) is the \( y \)-intercept, the point where the line crosses the \( y \)-axis.

The equation \( y = mx + b \) provides a powerful way to predict \( y \) for any given \( x \). It also makes graphing straightforward: starting from the \( y \)-intercept \( b \) and following the slope \( m \), you can define the entire line.

This form is particularly handy because it quickly translates into a visual graph, showing exactly where and how a line will pass through the coordinate plane. By understanding these components, you can efficiently analyze the positioning and inclination of any given line.