Problem 34 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(4x + 5y - 9 = 0\)

Step-by-Step Solution

Answer

The inclination of the line \(4x + 5y - 9 = 0\) is \(\arctan(-4/5)\) radians and \(\arctan(-4/5)*(180/π)\) degrees.

Step 1: Extract Coefficients

The first step is to identify the coefficients of x and y in the given equation. Therefore, from the equation \(4x + 5y - 9 = 0\), we can note that \(A = 4, B = 5\).

Step 2: Calculate Slope

We know that the slope of a line given in standard form \(Ax + By + C = 0\) is given by \(-A/B\). Therefore the slope of the given line will be \(-4/5\).

Step 3: Convert Slope to Radian Inclination

The tangent of the angle of inclination is equal to the slope of the line. To solve for the angle, one can take the inverse tangent ('arctan' or 'tan-1') of the calculated slope. Hence, we can calculate the inclination in radians by evaluating \(\arctan(-4/5)\) and use the standard radian to degree conversion to get the inclination in degrees by multiplying by 180/π.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It provides a clear and straightforward way to understand the line's behavior on a coordinate plane. By arranging the equation in the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept, it becomes easier to graph the line and predict its direction.

To convert the given line equation \(4x + 5y - 9 = 0\) into slope-intercept form, we'll solve for \(y\):

• First, isolate \(5y\): \(5y = -4x + 9\).
• Next, divide through by 5 to solve for \(y\): \(y = -\frac{4}{5}x + \frac{9}{5}\).

This result makes it clear that the slope, \(m\), is \(-\frac{4}{5}\), and the y-intercept, \(b\), is \(\frac{9}{5}\). With this form, it is easy to envision how the line travels downward due to its negative slope.

Angle of Inclination

The angle of inclination refers to the angle a line makes with the positive x-axis in a coordinate plane. This angle helps us understand how steep a line is. It is an important concept in coordinate geometry, indicating the orientation of the line.

The angle of inclination, \(\theta\), is determined by the slope \(m\) of the line. Specifically, \(\theta\) can be found using the formula \(\tan(\theta) = m\). This equation shows how the slope and the angle directly relate to each other. For our line, given a slope of \(-\frac{4}{5}\), \(\theta\) would be calculated using the inverse tangent.

The value for \(\theta\) will be negative because the line descends from left to right. This negative value is typical for angles measured counterclockwise from the positive x-axis when dealing with negative slopes.

Inverse Tangent

The inverse tangent function, often written as \(\arctan\) or \(\tan^{-1}\), is instrumental in finding the angle of inclination from a given slope. It is the inverse operation of the tangent function and returns the angle whose tangent is a specific value.

For our equation, the slope of the line is \(-\frac{4}{5}\). Therefore, the angle of inclination \(\theta\) is calculated as:

• \(\theta = \arctan\left(-\frac{4}{5}\right)\).

The result from this operation is an angle in radians. To convert to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

This conversion is necessary because angles can be more intuitive in degrees when applying concepts in real-world situations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using a coordinate system to study the geometric properties and relationships of figures within a plane. This branch is essential when dealing with equations of lines, angles, and distances, as it bridges algebra and geometry through the use of coordinates.

In our exercise, we explore how a linear equation, like \(4x + 5y - 9 = 0\), describes a line in a coordinate plane. By understanding the structure and elements of the equation:

• We can find the slope, which describes the line's steepness.
• We calculate the angle of inclination, explaining the line's angle with the x-axis.
• We employ inverse tangent functions, linking algebraic and geometric interpretations.

These tools within coordinate geometry allow for a comprehensive understanding of how lines behave and interact within the coordinate plane.