Problem 32 - Precalculus with Limits

Question

In Exercises 27-36, find the inclination \(\theta\) (in radians and degrees) of the line. \(-2\sqrt{3}x - 2y = 0\)

Step-by-Step Solution

Answer

The inclination of the line is \(\arctan(\sqrt{3})\) radians or \(60\) degrees.

Step 1: Rewrite the equation in slope-intercept form

Rearrange the equation \( -2\sqrt{3}x - 2y = 0 \) into slope-intercept form.\nDivide each term in the equation by \(-2\) to give: \(y = \sqrt{3}x\). This gives the slope \(m = \sqrt{3}\).

Step 2: Find the initial inclination in radians

Substitute the slope \(m\) into the arctangent function to find the initial inclination \(\theta\) in radians. This gives: \(\theta = \arctan(m) = \arctan(\sqrt{3})\).

Step 3: Convert the inclination from radians to degrees

To convert from radians to degrees, multiply the angle in radians by \(\frac{180}{\pi}\). This gives: \(\theta\text{ (degrees)} = \theta\text{ (radians)} \times \frac{180}{\pi}\). Substitute \(\theta\text{ (radians)} = \arctan(\sqrt{3})\) into this equation to find \(\theta\text{ (degrees)}\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is essential in algebra and calculus. It is one of the most common ways to write a linear equation. The standard slope-intercept form is expressed as :

\[ y = mx + b \]

where y is the dependent variable, x is the independent variable, m represents the slope of the line, and b is the y-intercept, which is the point where the line crosses the y-axis. Converting an equation like -2\textbackslash{}sqrt{3}x - 2y = 0 into slope-intercept form requires isolating y on one side. This makes it easier to visualize the line on a graph, as well as to identify the inclination and other characteristics of the line directly from the equation.

Slope of a Line

The slope of a line is a measure of its steepness and its direction. In the equation \[ y = \textbackslash{}sqrt{3}x \]

the slope is \( \textbackslash{}sqrt{3} \), which suggests that for every unit increase in x, y increases by \( \textbackslash{}sqrt{3} \) units. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend.

Calculating the inclination, or angle, of the line with respect to the x-axis involves trigonometry. The initial inclination \( \textbackslash{}theta \) in radians can be found using the arctangent (or inverse tangent) function, which is written as \( \textbackslash{}arctan(m) \) where m is the slope of the line. In trigonometry, arctangent is commonly used to retrieve the angle whose tangent is the given number, thus connecting the concepts of slope and angle.

Radians to Degrees Conversion

Angles can be measured in two common units: radians and degrees. In most mathematical contexts, radians are preferred, as they provide a direct link between the angle and the arc length in a circle. However, degrees are often more intuitive and are commonly used in everyday life.

The conversion between radians and degrees is based on the relationship that a full circle contains \( 2\textbackslash{}pi \) radians, which is equivalent to 360 degrees. Therefore, to convert an angle from radians to degrees, we use the formula:

\[ \text{degrees} = \text{radians} \times \frac{180}{\textbackslash{}pi} \]

For the given problem, where the inclination in radians is found to be \( \textbackslash{}arctan(\textbackslash{}sqrt{3}) \), we multiply this value by \( \frac{180}{\textbackslash{}pi} \) to obtain the angle in degrees. This conversion is vital for interpreting angles in a way that is more familiar and commonly used in various applications, such as navigation and land surveying.