Problem 30 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The inclination \( \theta \) of the line \( \sqrt{3}x - y + 2 = 0 \) is \( \frac{\pi}{3} \) radians or \( 60 \) degrees

Step 1: Convert into standard form

Rewrite the linear equation into the standard form \(y = mx+c\). By adjusting the equation given, you can get it as \(y = \sqrt{3}x - 2\)

Step 2: Identification of the slope

The coefficient of \(x\) in the standard form equation from step 1 gives the slope. So, here \(m = \sqrt{3}\)

Step 3: Calculation of the inclination in radians

Knowing that the inclination \( \theta \) is the arctangent of the slope \( m \), calculate it as \( \theta = \arctan(m) \). Substituting \( m = \sqrt{3} \) gives \( \theta = \arctan(\sqrt{3}) \). Simplifying this yields \( \theta = \frac{\pi}{3} \) radians

Step 4: Conversion of radians to degrees

To convert radians into degrees, multiply the radian measure by \(\frac{180}{\pi}\). So, \( \theta = \frac{\pi}{3} \times \frac{180}{\pi} \) simplifies to \( \theta = 60 \) degrees

Key Concepts

Linear equations

A linear equation is a mathematical expression that creates a straight line when plotted on a graph. It typically takes the form \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept, the point where the line crosses the y-axis.
Linear equations are foundational in mathematics because they model relationships with a constant rate of change. They are easy to visualize since each linear equation can be directly represented as a straight line on a Cartesian plane.

To work with linear equations, always aim to rearrange them into their standard form. This simple step helps in easily identifying the slope and y-intercept. Thus, it becomes straightforward to plot the line or compute its intersection with other lines.

Slope of a line

The slope of a line is a measure of its steepness, indicating whether it goes up or down as you move from left to right. It is denoted as \( m \) in the equation \( y = mx + c \).
The slope is calculated as the change in y-coordinate (rise) divided by the change in x-coordinate (run) between two points on the line. In mathematical terms, it is expressed as:

\[m = \frac{\Delta y}{\Delta x}\]

Finding the slope helps in understanding how the line behaves. A positive slope means the line ascends, while a negative slope indicates it descends. If the slope is zero, the line is perfectly horizontal. A larger absolute value of the slope denotes a steeper line.

In the context of inclination, the slope also assists in determining the angle of the line relative to a horizontal plane.

Radians and degrees conversion

Angles can be measured in radians or degrees, and converting between these units is often necessary in mathematics. The conversion hinges on knowing that \( 180\) degrees is equivalent to \( \pi \) radians.
To convert an angle from radians to degrees, you multiply the radian measurement by \( \frac{180}{\pi} \). Conversely, to go from degrees to radians, you multiply the degree measurement by \( \frac{\pi}{180} \).
As an example, to convert an angle of \( \frac{\pi}{3} \) radians to degrees, you compute:

\[\frac{\pi}{3} \times \frac{180}{\pi} = 60 \text{ degrees}\]

This conversion is crucial, especially in disciplines like trigonometry and calculus, where the choice of unit can influence the ease of computation and interpretation of results.