Problem 33 - Calculus with Concepts in Calculus

Question

Find the tangent of the angle \(\theta\) from the first line to the second line. \(y=4 x-2 ; 3 y=-2 x+7\)

Step-by-Step Solution

Answer

The tangent of the angle \(\theta\) is \(-\frac{14}{5}\).

Step 1: Identify the slope of the first line

The equation of the first line is given in the slope-intercept form, \(y = mx + c\), where \(m\) is the slope. For the line \(y = 4x - 2\), the slope \(m_1 = 4\).

Step 2: Reformat the second line to slope-intercept form

The equation of the second line is given as \(3y = -2x + 7\). To find the slope, rearrange it into slope-intercept form \(y = mx + c\). Divide every term by 3 to get \(y = -\frac{2}{3}x + \frac{7}{3}\). Thus, the slope \(m_2 = -\frac{2}{3}\).

Step 3: Calculate the tangent of the angle \(\theta\) between the two lines

The tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using the formula: \( \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \). Substitute \(m_1 = 4\) and \(m_2 = -\frac{2}{3}\) into the formula: \[\tan(\theta) = \frac{4 - \left(-\frac{2}{3}\right)}{1 + 4\left(-\frac{2}{3}\right)} = \frac{4 + \frac{2}{3}}{1 - \frac{8}{3}} = \frac{\frac{14}{3}}{-\frac{5}{3}} = -\frac{14}{5}.\]

Key Concepts

Tangent of an Angle

The tangent of an angle serves as a bridge between the geometric concept of an angle and the algebraic idea of a slope. When we talk about the angle between two lines, we are considering how steep one line is relative to the other. The tangent function captures this comparison by using the slopes of the lines involved.
The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is:

• \( \tan(\theta) = \frac{m_1 - m_2}{1 + m_1 m_2} \)

A positive tangent value indicates that the lines make an acute angle with each other. Meanwhile, a negative value signals an obtuse angle.
This formula arises from the trigonometric definition of tangent as the ratio of opposite over adjacent sides in a right triangle but is tuned to work with line slopes. Understanding this will help you visualize the correlation between angles in geometry and slopes in algebra.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Imagine a hill; the steeper the hill, the greater its slope. In mathematics, the slope is represented by the letter \(m\). It is calculated as the ratio of the vertical change to the horizontal change between two points on a line.

• The formula for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
• If a line is horizontal, its slope is zero; if vertical, the slope is undefined.

Identifying slopes is crucial for understanding the angle between lines as it indicates how each line tilts in relation to the horizontal axis.
Knowing how to find and use slopes gives you a solid foundation for delving deeper into geometry and calculus.

Slope-Intercept Form

The slope-intercept form of a line is a popular and convenient way to write and understand linear equations. Given by \( y = mx + c \), it clearly presents the slope \(m\) and the y-intercept \(c\) of the line.

• \(m\): Represents the slope or steepness of the line, showing how much \(y\) increases or decreases as \(x\) increases by 1 unit.
• \(c\): The y-intercept, which is the point where the line crosses the y-axis, that is where \(x = 0\).

Using the slope-intercept form allows for quick graphing of the line and deep insight into how changes in \(m\) and \(c\) affect the line's position and tilt.
Converting equations into this form can simplify complex problems, making it easier to analyze relationships between multiple lines by visually comparing their slopes and y-intercepts.