Problem 38 - Technical Mathematics with Calculus

Question

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=0$$

Step-by-Step Solution

Answer

0 degrees

Step 1: Understanding the Slope

The slope (m) of a line is a measure of its steepness, given by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. A slope of 0 indicates that there is no vertical change as you move along the line, meaning the line is perfectly horizontal.

Step 2: Relating Slope to Angle of Inclination

The angle of inclination of a line is the angle the line makes with the positive direction of the x-axis. A horizontal line makes a zero degree angle with the x-axis since it is perfectly aligned with it.

Step 3: Calculating the Angle of Inclination

Since the slope (m) is 0, indicating a horizontal line, the angle of inclination is also 0 degrees.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in algebra and geometry, representing the line's steepness and direction. It is usually denoted by the letter 'm' and is defined as the ratio of the 'rise' (the vertical change) to the 'run' (the horizontal change) between any two points on the line. The formula to find the slope is expressed as:
\[ m = \frac{\text{rise}}{\text{run}} \].
When the slope is positive, the line rises from left to right. Conversely, a negative slope means the line falls from left to right. If a line is moving horizontally without any rise or fall, it's said to have a slope of zero. This brings us to the concept of a horizontal line, which crucially affects the angle of inclination.

Horizontal Line

A horizontal line is a straight line that runs from left to right and is parallel to the x-axis in a Cartesian coordinate system. Because of its nature, the slope of a horizontal line is always 0. This is because the vertical change (rise) between any two points on a horizontal line is zero. Since slope is calculated based on the vertical change over the horizontal change, and the rise is zero, the slope formula simplifies to:
\[ m = \frac{0}{\text{run}} = 0 \].
A perfect horizontal line never ascends or descends, implying that there is no angle to the horizontal; this is key when we talk about the angle of inclination, as a zero slope leads to a zero angle.

Angle of Inclination Calculation

The angle of inclination refers to the angle formed between a line and the positive direction of the x-axis. Geometrically, it's the angle you would 'tilt' the x-axis to make it overlap with the line. This angle is particularly interesting because it provides a way to visualize the slope in a geometric context.
For most lines, the angle of inclination can be found by taking the arctangent (also known as the inverse tangent) of the slope:
\[ \text{Angle of inclination} = \arctan(m) \].
However, when it comes to a horizontal line with a slope of 0, the calculation simplifies significantly. Since the arctangent of 0 is 0, the angle of inclination for a horizontal line is:
\[ \text{Angle of inclination} = \arctan(0) = 0^\circ \]. This directly corresponds to the line being perfectly flat or 'level' relative to the x-axis. In the context of the example exercise, where the slope is given as 0, the line would have no tilt at all, and thus the angle of inclination would also be precisely 0 degrees.