Problem 44 - Precalculus with Limits

Question

In Exercises 37-46, find the angle \(\theta\) (in radians and degrees)between the lines. \(3x - 5y = 3\) \(3x + 5y = 12\)

Step-by-Step Solution

Answer

The angle \(\theta\) between the lines is \(Arctan(-\frac{15}{8})\) radians or \(Arctan(-\frac{15}{8}) \cdot \frac{180}{\pi}\) degrees

Step 1: Determine the slopes of the lines

Firstly, rearrange both equations in the form `y=mx+c` where `m` is the slope and `c` is the y-intercept. The slope \(m_1\) of the first line is found by re-arranging \(3x - 5y = 3\) which gives \(y_1 = \frac{3}{5}x - \frac{3}{5}\). Hence \(m_1 = \frac{3}{5}\).Similarly, for the second line \(3x + 5y = 12\), we rearrange to get \(y_2 = \frac{3}{5}x - \frac{12}{5}\) so \(m_2 = - \frac{3}{5}\).

Step 2: Calculate the angle

We can use the formula for the tangent of an angle between two lines: \[ \tan(\theta) = \frac{m_2 - m_1}{1 + m_1m_2} \]Substitute the calculated slopes:\[ \tan(\theta) = \frac{- \frac{3}{5} - \frac{3}{5}}{1 + \frac{3}{5}\cdot -\frac{3}{5}} = \frac{-\frac{6}{5}}{1 - \frac{9}{25}} = -\frac{6}{5} \cdot \frac{25}{16} = - \frac{15}{8} \]The next step is to take the arctangent of both sides. Using the restriction of the arctan function, we get\[ \theta = Arctan(-\frac{15}{8}) \].

Step 3: Convert from radians to degrees

Since the result from the arctan function is in radians, we have to convert it to degrees using the formula: \[ Degrees = Radians \cdot \frac{180}{\pi} \]Doing so gives us \(\theta = Arctan(-\frac{15}{8}) \cdot \frac{180}{\pi}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It is expressed as y = mx + c, where m stands for the slope of the line, and c is the y-intercept, which is the point where the line crosses the y-axis.

Understanding the slope is crucial since it tells us how steep the line is. A positive slope means the line is going uphill, while a negative slope indicates it's going downhill when moving from left to right. The slope of zero signifies a horizontal line, and an undefined slope (one that cannot be calculated because it would involve dividing by zero) corresponds to a vertical line.

In the case of our original problem, we rearranged the equation of each line to get them into the slope-intercept form so we could easily compare their slopes. This step is essential since the angle between two lines is directly related to the difference in their slopes.

Tangent of an Angle

The tangent of an angle in a right triangle is a ratio of the length of the opposite side to the length of the adjacent side. When dealing with lines, the tangent can also represent the slope of a line in a coordinate plane.

Now, when it comes to finding the angle between two intersecting lines, there's a specific formula that involves the tangent function: \( \tan(\theta) = \frac{m_2 - m_1}{1 + m_1m_2} \), where \( m_1 \) and \( m_2 \) are the slopes of the two lines, and \( \theta \) is the angle between the lines.

This formula allows us to see how the slope of each line influences the angle they form. A greater difference in the slopes will typically result in a larger angle between the lines.

Arctangent Function

The arctangent function, often written as ArcTan or Tan^-1, is the inverse of the tangent function. It allows us to find the angle whose tangent is a given number. This is particularly useful in trigonometry when we have the ratio of the sides of a right triangle and want to determine the angle itself.

After computing the tangent of the angle between two lines, we use the arctangent function to find the actual angle, \( \theta \). The arctangent function will return the angle in radians by default. It's worth noting that the range of the arctangent function is limited from \( -\frac{\text{π}}{2} \) to \( \frac{\text{π}}{2} \) (or from -90 degrees to 90 degrees), and this dictates the resulting angle should be within these bounds.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles. Since many real-world applications and even school problems generally use degrees, converting from radians to degrees is a common requirement.

To convert an angle given in radians to degrees, we use the conversion factor \( \frac{180}{\text{π}} \). The formula is given by multiplying the angle in radians by this conversion factor, which is approximately 57.2958. Specifically, \( \text{Degrees} = \text{Radians} \times \frac{180}{\text{π}} \).

In our exercise, after finding the angle in radians using the arctangent function, we needed to convert this angle into degrees to fully solve the problem. The ability to convert between radians and degrees allows students to work flexibly within various mathematical contexts and understand angles in both theoretical and practical settings.