Problem 34 - Calculus with Concepts in Calculus

Question

Find the tangent of the angle \(\theta\) from the first line to the second line. \(y=3 x+9 ; 4 y-11 x=6\)

Step-by-Step Solution

Answer

The tangent of the angle between the two lines is \(\frac{1}{37}\).

Step 1: Find Slopes of Both Lines

The first line is given in slope-intercept form: \(y = 3x + 9\). The slope of this line \(m_1\) is \(3\). Now, rearrange the second line into slope-intercept form by solving for \(y\). Start with the equation \(4y - 11x = 6\) and rearrange as follows: \[4y = 11x + 6\] \[y = \frac{11}{4}x + \frac{6}{4}\]. From this, the slope of the second line \(m_2\) is \(\frac{11}{4}\).

Step 2: Use Tangent of Angle Formula Between Two Lines

The tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by the formula: \[\tan(\theta) = \left|\frac{m_2 - m_1}{1 + m_1 \cdot m_2}\right|\]. Substituting the slopes from Step 1, the formula becomes: \[\tan(\theta) = \left|\frac{\frac{11}{4} - 3}{1 + 3 \cdot \frac{11}{4}}\right|\].

Step 3: Simplify and Calculate

First, simplify the numerator \(\frac{11}{4} - 3\): \[\frac{11}{4} - \frac{12}{4} = -\frac{1}{4}\]. Next, simplify the denominator \(1 + 3 \times \frac{11}{4}\): \[1 + \frac{33}{4} = \frac{4}{4} + \frac{33}{4} = \frac{37}{4}\]. Therefore, \[\tan(\theta) = \left|\frac{-\frac{1}{4}}{\frac{37}{4}}\right| = \left|-\frac{1}{4} \times \frac{4}{37}\right| = \frac{1}{37}\].

Step 4: Conclusion

The tangent of the angle \(\theta\) between the two lines is \(\frac{1}{37}\), which represents the measure of how the first line deviates from being parallel to the second line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line so that it is easy to understand and work with. It looks like this: \[y = mx + b\]Where:

• \(m\) is the slope of the line. The slope tells you how steep the line is. A larger slope means a steeper line.
• \(b\) is the y-intercept. This is the point where the line crosses the y-axis.

Understanding this form is useful when you need to quickly find the slope and the y-intercept of a line. For example, the line \(y = 3x + 9\) is already in slope-intercept form:

• The slope \(m_1\) is 3, indicating the line rises 3 units for each unit it moves to the right.
• The y-intercept is 9, which means the line crosses the y-axis at the point \((0, 9)\).

Recognizing this form helps solve more complex geometric problems, including finding angles between two lines.

Equation of a Line

The equation of a line is a mathematical statement that describes all the points that lie on the line. There are various formats for representing this equation, with the slope-intercept form being one of the most common. However, when a line isn't initially presented in slope-intercept form, like the line given by the equation \[4y - 11x = 6\],we often need to rearrange it to find the slope. Here's how to do it:

• First, solve for \(y\) to rewrite the equation in slope-intercept form.
• Starting from \(4y - 11x = 6\), add \(11x\) to both sides:
  \[4y = 11x + 6\]
• Then, divide everything by 4 to isolate \(y\):
  \[y = \frac{11}{4}x + \frac{6}{4}\]

This rearrangement shows that the slope \(m_2\) of the second line is \(\frac{11}{4}\). Solving the equation of a line to this form makes identifying the slope straightforward, which is crucial for calculating angles, among other applications.

Tangent Function

The tangent function is a trigonometric function that relates the angle between two lines to the slopes of those lines. When you want to find out how much one line deviates from another, the tangent of the angle between them helps. This is done using the slopes from each line's equation.The formula to find the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is:\[\tan(\theta) = \left|\frac{m_2 - m_1}{1 + m_1 \cdot m_2}\right|\]Here's how you use the formula with our example:

• Substitute the slopes \(m_1 = 3\) and \(m_2 = \frac{11}{4}\) into the formula.
• Calculate the values in the formula:
  \[\tan(\theta) = \left|\frac{\frac{11}{4} - 3}{1 + 3 \cdot \frac{11}{4}}\right|\]

Simplifying gives you \(\tan(\theta) = \frac{1}{37}\), which shows that the angle between the lines is relatively small. Understanding the tangent function allows us to determine the angle's measure, conveying how steeply one line diverges from another.