Problem 44 - Precalculus: A Concise Course

Question

Find the angle \(\alpha\) between two nonvertical lines \(L_{1}\) and \(L_{2}\). The angle \(\alpha\) satisfies the equation $$\tan \boldsymbol{\alpha}=\left|\frac{m_{2}-m_{1}}{1+m_{2} m_{1}}\right|$$ where \(m_{1}\) and \(m_{2}\) are the slopes of \(L_{1}\) and \(L_{2}\), respectively. (Assume that \(\left.m_{1} m_{2} \neq-1 .\right)\) $$ \begin{array}{l} L_{1}: 2 x-y=8 \\ L_{2}: \quad x-5 y=-4 \end{array} $$

Step-by-Step Solution

Answer

The angle \(\alpha\) between the two lines is approximately \(52.13^\circ\).

Step 1: Find the Slope of Line \(L_1\)

Rewrite the equation of line \(L_1 (2x - y = 8)\) in the slope-intercept form (y = mx+b). Subtract \(2x\) from both sides to get \(-y = -2x + 8\). Then, multiply through by -1 to get \(y = 2x - 8\). Now, it can be seen that the slope \(m_1\) of line \(L_1\) is 2.

Step 2: Find the Slope of Line \(L_2\)

Rewrite the equation of line \(L_2 (x - 5y = -4)\) in the slope-intercept form (y = mx+b). Subtract \(x\) from both sides to get \(-5y = -x - 4\). Then, divide through by -5 to get \(y = 0.2x + 0.8\). Now, it can be seen that the slope \(m_2\) of line \(L_2\) is 0.2.

Step 3: Calculate the Angle

Now that we have the slopes \(m_1\) and \(m_2\), the angle can be calculated using the given formula. The formula is: \(\tan(\alpha) = | \frac {m_2 - m_1} {1+m_1 \cdot m_2} |\). So substituting the values of \(m_1\) and \(m_2\) we get: \(\tan(\alpha) = \left| \frac {0.2 - 2} {1 + (2 \cdot 0.2)} \right| = |-1.8/1.4 | = 1.28571\). To obtain the angle \(\alpha\), we calculate the arctan of 1.28571, and we get approximately \(52.13^\circ\).

Step 4: Results Verification

The problem assumed that \( m_1 \cdot m_2 \neq -1 \), so let's verify it with our result. Substituting \(m_1 = 2\) and \(m_2 = 0.2\) in \(m_1 \cdot m_2 = -1\), we get \(2*0.2 = 0.4 \neq -1\), confirming our solution is correct.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is crucial for solving problems related to angles between lines. The slope-intercept form is expressed as:

• \( y = mx + b \)
• Where \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The primary advantage of the slope-intercept form is how it directly shows the slope, which signifies the tilt or inclination of the line. To rewrite any linear equation in this form, solve for \( y \) so it stands alone on one side of the equation. For example, consider line \( L_1: 2x - y = 8 \). By rearranging terms, we find its slope is \( m_1 = 2 \). Similarly, line \( L_2: x - 5y = -4 \) has a slope of \( m_2 = 0.2 \) after conversion to the slope-intercept form. This approach simplifies finding slopes and ultimately helps determine angles between lines, which all relate to their slopes.

Tangent of Angle

When we discuss the angle between two lines, the tangent function comes into play. The tangent of an angle, denoted as \( \tan(\alpha) \), connects the slopes of two intersecting lines. For two lines with slopes \( m_1 \) and \( m_2 \), the tangent of the angle \( \alpha \) between them is found using the formula:

• \( \tan(\alpha) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \)

The absolute value ensures the angle is non-negative. The formula gives the tangent value of the angle, which signifies how steep the angle is formed by these crossing lines. In practical terms, this tells us how much one line inclines with respect to the other. Calculating the tangent precisely helps us later find the real angle using further mathematical functions like the arctan.

Arctan Function

To find the actual angle when we have its tangent, we use the arctan (or inverse tangent) function. This function essentially works in reverse of the tangent, transforming a tangent value back into an angle.

• The arctan is usually denoted as \( \arctan \) or \( \tan^{-1} \).
• It converts the result back into degree or radian measure, giving the smallest angle formed by two lines.

For instance, if you have computed \( \tan(\alpha) = 1.28571 \), you would apply the arctan function to find \( \alpha \). The result is \( \alpha \approx 52.13^\circ \). Arctan is essential because it translates a straightforward numerical calculation into understanding the actual geometric context. It's a step that bridges computation with a clear visualization of the angle's magnitude.

Slope of a Line

The slope of a line reflects how steep the line is and is a key factor in many geometric calculations, including finding the angle between lines. Simply put, the slope \( m \) is the change in \( y \) over the change in \( x \) - mathematically, it is represented as:

• \( m = \frac{\Delta y}{\Delta x} \)

This means for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The slope provides a numerical value to express the line's angle relative to the horizontal. When we know the slope of two lines, calculating the tangent of the angle between them becomes straightforward. This understanding allows us to use the slope-intercept form effectively and is foundational for finding angles between intersecting lines.