Problem 41 - Precalculus

Question

Find the inclination \(\theta\) (in radians and degrees) of the line. $$6 x-2 y+8=0$$

Step-by-Step Solution

Answer

The inclination of the line \(6x - 2y + 8 = 0\) is \(arctan(3)\) radians or \(arctan(3) * \frac{180}{\pi}\) degrees.

Step 1: Equation Transformation

Re-arrange the given equation \(6x - 2y + 8 = 0\) into slope-intercept form \(y = mx + c\). After doing so, we get the equation \(y = 3x + 4\)

Step 2: Find the Slope

The slope 'm' of the equation \(y = 3x + 4\) is 3, which means the tangent of the inclination angle \(\theta\) is 3.

Step 3: Calculate Inclination in Radians

The inclination \(\theta\) is the arctangent of the slope. So, \(\theta = arctan(m) = arctan(3)\)

Step 4: Calculate Inclination in Degrees

Convert the inclination from radians to degrees using the conversion factor \(180/\pi\). So, \(\theta(in degrees) = \theta(in radians) * \frac{180}{\pi}\)

Key Concepts

Slope-Intercept Form

In algebra, the equations of lines are often expressed in different forms to make understanding them easier. One of the most common forms is the slope-intercept form. This is represented by the equation \(y = mx + c\), where:

• \(m\) is the slope of the line.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

To better understand how a line behaves, we can rearrange any linear equation into this form. For example, if we have the equation \(6x - 2y + 8 = 0\), transforming it into the slope-intercept form involves rearranging it as follows:

• Firstly, isolate \(y\) by subtracting \(6x + 8\) from both sides to get \(-2y = -6x - 8\).
• Divide every term by \(-2\), resulting in \(y = 3x + 4\).

Now the slope \(m\) is 3, showing us the line rises 3 units for every 1 unit it moves to the right.

Arctangent Function

Understanding the inclination of a line often involves trigonometry, especially when dealing with angles. The arctangent function \(arctan\) is crucial here. It helps find the angle by which a line is inclined with respect to the x-axis.In mathematical terms:

• The function \(arctan(m)\) gives the angle (inclination \(\theta\)) whose tangent is the slope \(m\) of the line.

Given the slope \(m = 3\) from the line equation \(y = 3x + 4\), you can find the inclination \(\theta\) in radians by calculating \(\theta = arctan(3)\). The value derived here is the smallest positive angle between the line and the x-axis.
This output in radians gives a detailed sense of how steep or flat the line is visually as seen on a graph.

Conversion Between Radians and Degrees

Angles can be measured in multiple units, mainly radians and degrees. Though often interchanged, understanding how to convert between them is a key mathematical skill. The conversion between radians and degrees is based on the circumference of a circle, with key constants being:

• A full circle is \(2\pi\) radians or 360 degrees.
• Thus, \(\pi\) radians are equal to 180 degrees.

To convert a given angle from radians to degrees, you multiply by the factor \(\frac{180}{\pi}\). For the inclination \(\theta\) calculated as \(arctan(3)\) in radians, convert it to degrees by using:
\(\theta(\text{in degrees}) = \theta(\text{in radians}) \times \frac{180}{\pi}\)
This calculation helps understand angles in terms commonly used in geometry and trigonometry, making them friendlier for interpretation in everyday contexts.