Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(\ell_{1}\) and \(\ell_{2}\) be two nonvertical intersecting lines with slopes \(m_{1}\) and \(m_{2}\), respectively. If \(\theta\), the angle from \(\ell_{1}\) to \(\ell_{2}\), is not a right angle, then $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$

Step-by-Step Solution

Answer

The formula \( \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \) applies if \( \theta \) is not a right angle and lines are non-vertical.

Step 1: Understand the Formula for the Angle Between Two Lines

To find the angle \( \theta \) between two intersecting lines with slopes \( m_1 \) and \( m_2 \), we use the formula \( \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \). This formula is applicable when the angle is not a right angle.

Step 2: Apply the Formula to the Problem

We have line \( \ell_1 \) with slope \( m_1 \) and line \( \ell_2 \) with slope \( m_2 \). The angle \( \theta \) formed between these two lines, which is not right-angled, can be calculated using the formula directly as \( \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \).

Step 3: Check Conditions for Validity of the Formula

The formula \( \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \) is valid as long as neither of the lines is vertical (ensured by slopes being finite) and the angle \( \theta \) is not a right angle. The lines \( \ell_1 \) and \( \ell_2 \) are non-vertical, and the problem states that \( \theta \) is not a right angle.

Step 4: Conclude the Solution

Given all conditions are satisfied, we can conclude that the expression \( \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \) correctly calculates the tangent of the angle \( \theta \) between the two intersecting lines \( \ell_1 \) and \( \ell_2 \).

Key Concepts

Slopes of Lines

The slope of a line is a crucial concept in geometry and algebra. It represents the steepness or the incline of a line, expressed as a ratio of the vertical change to the horizontal change between two distinct points on the line. This is often described with the letter "m." For example, if you have two points on a line,

• (x₁, y₁) and (x₂, y₂),

then the slope \(m\) is calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]A positive slope means the line inclines upwards as it moves from left to right, whereas a negative slope indicates a decline. Zero slope describes a horizontal line, and an undefined slope corresponds typically to a vertical line. Understanding slopes is fundamental when calculating the angle between two non-vertical intersecting lines, as each line's slope is a component in this derivation.

Tangent of Angle

In trigonometry, the tangent of an angle is a function that relates the angle

• to the ratio of the opposite side to the adjacent side in a right-angled triangle.

When dealing with non-perpendicular intersecting lines, we use this concept to determine the angle between them. The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by:\[\tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2}\]This formula comes handy when we do not want to calculate the angle directly but instead need its tangent value. It's important to ensure that the lines are not forming a right angle, as this would make the formula invalid due to division by zero when \(m_1 m_2 = -1\) which would make the denominator zero. Thus, knowing how tangent functions operate helps in understanding these conditions and applying the formula effectively.

Non-Vertical Lines

Non-vertical lines are simply lines that have a well-defined slope that is not infinite. Vertical lines, on the other hand, have undefined slopes because their vertical change is infinite compared to any horizontal change (zero in this case). For two lines to use the formula for finding the angle mentioned above, they must not be vertical, which allows the slopes \(m_1\) and \(m_2\) to be finite. This condition ensures continuity and avoids undefined mathematical operations in calculations like

• division by zero.

Non-vertical lines can contain any positive, negative, or zero value for their slopes. This property is crucial when applying the formula for finding the tangent of the angle between lines, as having a non-vertical line guarantees the formula remains valid. Identifying non-vertical lines is thus important for successful and accurate computation of angles in geometric problems.