Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

The angle of inclination \(\alpha\) of a line is the smallest positive angle from the positive \(x\)-axis to the line ( \(\alpha=0\) for a horizontal line). Show that the slope \(m\) of the line is equal to \(\tan \alpha\).

Step-by-Step Solution

Answer

The slope \( m \) of a line inclined at angle \( \alpha \) is \( m = \tan \alpha \).

Step 1: Define the slope of the line

The slope \( m \) of a line is defined as the ratio of the change in \( y \) to the change in \( x \). If the line makes an angle \( \alpha \) with the positive \( x \)-axis, the slope is given by \( m = \frac{\Delta y}{\Delta x} \).

Step 2: Understand the meaning of tangent of an angle

The tangent of an angle \( \alpha \), denoted as \( \tan \alpha \), is defined as the ratio of the opposite side to the adjacent side in a right triangle. In the context of a line with angle \( \alpha \) to the positive \( x \)-axis, this becomes the ratio \( \frac{\text{vertical change}}{\text{horizontal change}} \).

Step 3: Relate the definitions of slope and tangent

For a line inclined at an angle \( \alpha \), the slope \( m \) is given by \( m = \frac{\Delta y}{\Delta x} \). The angle \( \alpha \) by definition means the tangent of that angle \( \tan \alpha = \frac{\Delta y}{\Delta x} \). Thus, \( m = \tan \alpha \).

Step 4: Conclude with the equality of slope and tangent

By the definition of slope and the trigonometric property involving the tangent of the angle of inclination, we have shown that \( m = \tan \alpha \). Therefore, the slope of the line is equal to \( \tan \alpha \).

Key Concepts

Angle of Inclination

The angle of inclination is a fundamental concept in understanding the orientation of a line on a coordinate plane. It represents the smallest positive angle that a line makes with the positive direction of the x-axis. This angle, usually denoted by \( \alpha \), is measured in a counterclockwise direction and provides a geometric interpretation of how steep or flat a line is.
For a horizontal line, this angle is \( \alpha = 0 \), because it lies entirely along the x-axis. As the line tilts upwards, its angle of inclination increases, approaching \( \alpha = 90^\circ \) for a perfectly vertical line. Understanding this angle helps in connecting geometric angles with algebraic properties like the slope of the line.

Slope of a Line

The slope of a line is a crucial algebraic concept used to describe the direction and steepness of the line. Mathematically, it is represented as \( m \) and calculated as the ratio of the change in the y-coordinate (vertical change) to the change in the x-coordinate (horizontal change). This can be expressed as \( m = \frac{\Delta y}{\Delta x} \).

• When the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it goes from left to right.
• A zero slope signifies a horizontal line, indicating no vertical change.
• An undefined slope corresponds to a vertical line, where there is no horizontal change.

Understanding how the slope connects to the line's inclination helps in grasping its geometric behavior.

Tangent Function

The tangent function is a key trigonometric function relevant to calculus and geometry. It relates to angles in right triangles and is defined as the ratio of the opposite side to the adjacent side, observed within the triangle. For an angle \( \alpha \) formed between a line and the positive x-axis, this ratio is described by \( \tan \alpha = \frac{\text{opposite side}}{\text{adjacent side}} \), which aligns with how we describe slopes.
In the context of lines on a Cartesian plane, the tangent of the angle \( \alpha \) at which a line inclines from the x-axis equals the slope. This provides a direct relationship between a geometric angle and its algebraic representation as a slope, thus bridging trigonometric functions and linear equations.

Trigonometry in Calculus

Trigonometry plays a significant role in calculus by aiding in the analysis of angles and slopes on graphs. Through trigonometric functions like tangent, we can connect the angle of inclination of a line to its slope, allowing for a deeper understanding of curves and graphical data.
Integral and differential calculus utilize these principles:

• Integrals often require trigonometric identities to simplify and evaluate areas under curves.
• In differentiation, understanding how angles relate to slopes assists in finding derivatives of functions, especially when considering tangent or normal lines to curves.

These trigonometry concepts provide essential tools for solving real-world problems involving angles, optimizing functions, and analyzing motion within calculus.