Question

Find the angle of inclination of the following lines (see Problem 38 ). (a) \(y=\sqrt{3} x-7\) (b) \(\sqrt{3} x+3 y=6\)

Short answer

Line (a) has an angle of inclination of \(60^\circ\), and line (b) has \(-30^\circ\) or \(330^\circ\).

Step-by-step solution

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + c\), where \(m\) is the slope. The angle of inclination \(\theta\) of a line is related to its slope \(m\), with \(\theta = \tan^{-1}(m)\). Here, the problem is to find the angle of inclination of the provided lines.

Determine the Slope of Line (a)

For line (a), the equation is already in slope-intercept form: \(y = \sqrt{3}x - 7\). Therefore, the slope \(m\) is \(\sqrt{3}\).

Calculate the Angle of Inclination for Line (a)

Use the formula \(\theta = \tan^{-1}(m)\) to calculate the angle of inclination for line (a). Substitute \(m = \sqrt{3}\), so \(\theta = \tan^{-1}(\sqrt{3})\). The angle \(\theta\) is \(60^\circ\) since \(\tan(60^\circ) = \sqrt{3}\).

Rewrite Line (b) in Slope-Intercept Form

Line (b) is given as \(\sqrt{3}x + 3y = 6\). Solve for \(y\) to put it in slope-intercept form: \(3y = -\sqrt{3}x + 6\) and then \(y = -\frac{\sqrt{3}}{3}x + 2\). Now the slope \(m\) is \(-\frac{\sqrt{3}}{3}\).

Calculate the Angle of Inclination for Line (b)

Using the slope \(m = -\frac{\sqrt{3}}{3}\), compute \(\theta = \tan^{-1}(-\frac{\sqrt{3}}{3})\). This corresponds to an angle of \(-30^\circ\) (in standard position, this equates to an angle of \(330^\circ\)).

Key concepts

Angle of Inclination

The angle of inclination is a fundamental concept in calculus and geometry which describes the angle a line makes with the positive x-axis. This angle is usually measured in degrees or radians. By convention, angles are measured counterclockwise from the positive x-axis. The angle of inclination, often denoted as \( \theta \), gives us a visual representation of the steepness or direction of a line.
It's important to remember that the angle of inclination is restricted to values between \(0^\circ\) and \(180^\circ\). Here's how it works in simple terms:

• If a line is rising to the right, the angle will be between \(0^\circ\) and \(90^\circ\).
• If the line is horizontal, the angle is \(0^\circ\).
• If a line falls to the right, the angle is between \(90^\circ\) and \(180^\circ\).

To find the angle of inclination, we use the relationship \( \theta = \tan^{-1}(m) \), where \( m \) is the slope of the line. This formula allows us to find the angle if we know the slope.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to represent the equation of a line. This form is given as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept. The y-intercept \( c \) shows where the line crosses the y-axis.
Understanding this form is crucial because it enables us to quickly identify the slope and y-intercept from any linear equation. For example:

• The equation \( y = 2x + 3 \) has a slope \( m = 2 \) and y-intercept \( c = 3 \).
• The equation \( y = -\frac{1}{2}x + 4 \) reveals a slope \( m = -\frac{1}{2} \) and y-intercept \( c = 4 \).

Using the slope-intercept form makes it easy to plot lines on a graph and check their inclination. It offers a straightforward way to transition from a line's equation to finding its slope and calculating other properties.

Trigonometric Functions

Trigonometric functions are mathematical relationships involving angles and the ratios of a triangle’s sides. In the context of a line’s angle of inclination, the tangent function is most relevant. For any angle \( \theta \), the tangent function establishes the ratio of the opposite side to the adjacent side in a right triangle.

• The tangent of an angle \( \theta \) is calculated as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• To find an angle from a slope, the inverse tangent or arctangent function, denoted as \( \tan^{-1} \), is used, which gives the angle \( \theta = \tan^{-1}(m) \).

By using these trigonometric relationships, we can compute angles from slopes, an essential step in determining a line’s angle of inclination. Understanding trigonometric functions is valuable not just for this purpose, but also across various branches of mathematics.

Slope of a Line

The slope of a line is a measure of its steepness or direction. It is calculated as the "rise over run"—the change in the y-coordinates divided by the change in the x-coordinates for any two points on the line. In formula terms, if a line passes through points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]A few key points about the slope of a line:

• A positive slope means the line is ascending as it moves from left to right.
• A zero slope indicates a horizontal line.
• A negative slope signifies the line is descending as it moves from left to right.
• Vertical lines have an undefined slope as their x-coordinates are constant.

Understanding slope is crucial because it directly influences the calculation of the angle of inclination. It provides insight into the line’s behavior in a coordinate plane, shaping our understanding of both visual and algebraic representations.