Problem 19 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line with slope \(m\). $$m=1$$

Step-by-Step Solution

Answer

The line has an inclination of \(\pi/4\) radians or 45 degrees.

Step 1: Understand the relationship between the slope and the angle of inclination

The slope \(m\) of a line and its angle of inclination \(\theta\) are related through the tangent function: \(m = \tan(\theta)\). Here, \(m\) indicates the steepness of the line and \(\theta\) represents the counterclockwise angle it makes with the x-axis.

Step 2: Find the angle in radians

Given that \(m=1\), you can find the angle \(\theta\) by calculating the inverse tangent of 1. This can be written as \(\theta = \arctan(m)\) or \(\theta = \arctan(1)\). Because the tangent of \(\pi/4\) or 45 degrees is 1, the arctan of 1 is \(\pi/4\) radians.

Step 3: Convert from radians to degrees

To convert radians to degrees, multiply the radian measure by \(180/\pi\). Therefore, \(\theta = (\pi/4) * (180/\pi) = 45\) degrees. This means the line makes a 45 degree angle with the x-axis.

Key Concepts

Slope of a Line

The slope of a line, often represented by \( m \), is a measure of the line's steepness. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula is given by:\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]This ratio helps us understand how much the line rises for every unit it moves horizontally. A positive slope indicates that the line is ascending, while a negative slope signifies a descending line.
For example:

• A slope of \( m = 1 \) means that for every unit the line moves right, it also moves up one unit, indicating a 45-degree angle with the x-axis.
• A slope of \( m = 0 \) corresponds to a horizontal line.

Understanding the slope is vital for determining the angle of inclination, which is the angle formed between the line and the positive x-axis.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in calculating angles when given trigonometric ratios. They are the "reverse" of regular trigonometric functions, linking an angle to a given trigonometric value. When the slope \( m \) of a line is known, the angle of inclination \( \theta \) can be determined using the inverse tangent function, or \( \arctan \). The equation \( m = \tan(\theta) \) reveals that:\[ \theta = \arctan(m) \]This expresses that \( \theta \) is the angle whose tangent is \( m \).
For a slope of 1, since \( \tan(\pi/4) = 1 \), we have that \( \arctan(1) = \pi/4 \) radians.
These functions are invaluable in fields like physics, engineering, and navigation, where angles derived from slopes are part of problem-solving.

Radian to Degree Conversion

Radians and degrees are two units for measuring angles. Radians are often used in calculus and higher mathematics, while degrees are used in everyday settings, like construction and geography. Converting between these two can help bridge practical understanding with mathematical principles. Here's how you can convert:To convert from radians to degrees, the formula used is:\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]This formula arises because a full circle is \( 2\pi \) radians or 360 degrees. Thus, \( 1\) radian equals about \( 57.2958 \) degrees.
For example, the angle \( \pi/4 \) radians equals \( 45\) degrees, because:\[ \frac{\pi}{4} \times \frac{180}{\pi} = 45 \] Degrees offer a familiar way to relate angle measurements to everyday experiences, while radians provide a more natural approach in advanced mathematical calculations.