Problem 41 - Precalculus with Limits

Question

In Exercises 37-46, find the angle \(\theta\) (in radians and degrees)between the lines. \(x - 2y = 7\) \(6x + 2y = 5\)

Step-by-Step Solution

Answer

The angle \(\theta\) between the lines \(x - 2y = 7\) and \(6x + 2y = 5\) is approximately \(1.2\) radians or \(68.2\) degrees.

Step 1: Calculate the Slopes

First, find the slopes of the given lines using the formula \(-a/b\). For the first line (where \(a = 1\) and \(b = -2\)), the slope is \(1/2\). For the second line (where \(a = 6\) and \(b = 2\)), the slope is \(6/2 = 3\). So \(m_1 = 1/2\) and \(m_2 = 3\).

Step 2: Find the Tangent of the Angle

Next, substitute \(m_1\) and \(m_2\) in the formula for the tangent of the angle between two lines. The tangent of the angle \(\theta\) is given by \( | (m_2 - m_1) / (1 + m_1*m_2) | \). After you substitute the known values, you get \( | (3 - 1/2) / (1 + 1/2*3) | \) which simplifies to \(5/2\). So, \(\tan(\theta) = 5/2\).

Step 3: Calculate the Angle

The next step is to find the angle \(\theta\). If \(\tan(\theta) = 5/2\), then \(\theta = \arctan(5/2)\) in radians. And since there are \(180/\pi\) degrees in each radian, the angle in degrees is \( \arctan(5/2) * 180/\pi \). This operation results in \(\theta \approx 1.19\) radians or \(\theta \approx 68.2\) degrees.

Step 4: Round the Results

Finally, round the calculated angle to the nearest tenth to get more understandable results: \(\theta \approx 1.2\) radians and \(\theta \approx 68.2\) degrees.

Key Concepts

Radians and Degrees

Understanding radians and degrees is key in measuring angles. They are two different units for the same measurement. A full circle is 360 degrees, but in radians, it is measured as \(2\pi\). This means:

• 180 degrees = \(\pi\) radians
• 1 degree = \(\pi/180\) radians
• 1 radian = 180/\pi\ degrees

The exercise asks us to find the angle between two lines in both radians and degrees. This conversion is simple once you know these relationships, allowing for easier communication and understanding of angle measures in various contexts."

Slopes of Lines

The slope of a line measures how steep it is. It’s the ratio of the vertical change to the horizontal change between two points on the line. For a linear equation in the form \(ax + by = c\), the slope \(m\) is found using \(-a/b\). In our exercise:

• For \(x - 2y = 7\), slope \(m_1 = 1/2\)
• For \(6x + 2y = 5\), slope \(m_2 = 3\)

These values allow us to analyze how these lines rise and run, crucial for the next steps of finding the angle between them."

Tangent of Angle

The tangent function, from trigonometry, helps relate the slopes to the angle between the lines. For two lines with slopes \(m_1\) and \(m_2\), the formula for the tangent of the angle \(\theta\) between them is:\[\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 \cdot m_2} \right|\]This formula considers the difference in slopes and how they interact. For our lines:

• \(m_1 = 1/2\)
• \(m_2 = 3\)

Substituting in gives \(\tan(\theta) = 5/2\). The tangent ratio provides a tangible step toward finding the actual angle."

Arctangent

Arctangent, often written as \(\arctan\), is the inverse of the tangent function. It helps us find the angle itself once we know the tangent. If \(\tan(\theta) = 5/2\), then \(\theta = \arctan(5/2)\). This result is often initially in radians. To convert it into degrees, multiply by \(180/\pi\).Using a calculator, we find:

• \(\theta \approx 1.19\) radians
• \(\theta \approx 68.2\) degrees

The arctangent function is invaluable in transitioning from a numerical output to an understandable angle."

Formula for Angle Between Lines

To find the angle between two lines, we use a specific formula involving their slopes. The final formula to remember is:\[\theta = \arctan\left(\left| \frac{m_2 - m_1}{1 + m_1 \cdot m_2} \right|\right)\]This formula effectively translates the relationship of line inclines into an angle measure.

• Substitute known slopes to find \(\tan(\theta)\)
• Use \(\arctan\) to find \(\theta\) in radians
• Convert to degrees if needed

This method gives us a comprehensive understanding of how the two lines relate to each other in space."