Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the angle \(\alpha\) between the two nonvertical lines \(L_{1}\) and \(L_{2}\) (assume \(L_{1}\) and \(L_{2}\) are not perpendicular). The angle \(\alpha\) satisfies the equation \(\tan \alpha=\left|\frac{m_{2}-m_{1}}{1+m_{2} m_{1}}\right|\) where \(m_{1}\) and \(m_{2}\) are the slopes of \(L_{1}\) and \(L_{2}\), respectively. $$\begin{aligned} &L_{1}: 2 x+y=8\\\ &L_{2}: \quad x-5 y=-4 \end{aligned}$$

Step-by-Step Solution

Answer

The angle \( \alpha \) between the two lines is approximately 74.49 degrees.

Step 1: Calculate the slope of each line

The slope \( m \) of a line defined by the equation \( ax + by = c \) can be calculated by \( m = -a/b \). Here, for line \( L_1: 2x + y = 8 \), the slope \( m_1 \) is given by \( -2/1 = -2 \). For the line \( L_2: x - 5y = -4 \), the slope \( m_2 \) is given by \( -1/-5 = 1/5 \).

Step 2: Substitute the slopes in the formula

The tangent of the angle between the two lines is calculated using the formula \( \tan \alpha = |(m_2 - m_1) / (1 + m_2 m_1)| \). Substituting \( m_1 = -2 \) and \( m_2 = 1/5 \), we get \( \tan \alpha = |(1/5 - (-2)) / (1 + 1/5*(-2))| = |2.2 / 0.6| = 3.67.

Step 3: Calculate the angle

To find the angle \( \alpha \) between the lines, use the inverse tangent function. Therefore, \( \alpha = \arctan(3.67) = 74.49 \) degrees when rounded to two decimal places.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness or incline. It tells us how much the line rises (or falls) as it moves horizontally. In formula terms, if you have a line expressed in the standard form of a linear equation, like \( ax + by = c \), you can find its slope using the formula \( m = \frac{-a}{b} \). Here, the values of \( a \) and \( b \) come directly from the equation's coefficients.

For example, take the equation for line \( L_1: 2x + y = 8 \). Here, \( a = 2 \) and \( b = 1 \), so the slope \( m_1 \) is calculated as \(-2/1\). This gives a slope of \(-2\), indicating that the line falls two units for every unit it moves to the right.

For line \( L_2: x - 5y = -4 \), we use \( a = 1 \) and \( b = -5 \). Thus, the slope \( m_2 \) is \(-1/-5\), which simplifies to \( 1/5 \), meaning the line rises slowly, increasing only one unit upwards for every five units it moves to the right.

Tangent of an Angle

The tangent of an angle in a right triangle is a basic trigonometric function. It is a ratio comparing the length of the opposite side to the length of the adjacent side. In the context of angles between lines, the tangent function helps us translate slopes into angle measures.

This is crucial when trying to find the angle between two lines because the difference in their slopes gives information about how the lines interact at their intersection point. The formula \( \tan \alpha = \left| \frac{m_2 - m_1}{1 + m_2 m_1} \right| \) is derived from the tangent function to express the tangent of the angle \( \alpha \) between two lines with slopes \( m_1 \) and \( m_2 \).

In the given problem, substituting \( m_1 = -2 \) and \( m_2 = 1/5 \) into the formula gives: \( \tan \alpha = \left| \frac{1/5 - (-2)}{1 + (1/5)\times(-2)} \right| = \left| \frac{2.2}{0.6} \right| = 3.67 \). Thus, \( \tan \alpha \) provides a direct path to finding the actual angle between lines.

Inverse Tangent Function

The inverse tangent function, also known as arctangent or \( \arctan \), reverses the tangent process. If you know the tangent of an angle, you can find the degree measure of that angle using \( \arctan \).

Using the \( \arctan \) function is straightforward. Once you have \( \tan \alpha \), as calculated from the slopes, \( \alpha \) can be found through \( \alpha = \arctan(\tan \alpha) \). For example, if \( \tan \alpha = 3.67 \), then \( \alpha = \arctan(3.67) \).

This computation gives the value \( \alpha = 74.49 \) degrees (when rounded), indicating the angle formed by the intersection of the two lines. The inverse tangent function helps convert the abstract slope relationship back into a more intuitive concept, like degrees, which is easier to understand and visualize.