Problem 43 - Precalculus with Limits

Question

In Exercises 37-46, find the angle \(\theta\) (in radians and degrees)between the lines. \(x + 2y = 8\) \(x - 2y = 2\)

Step-by-Step Solution

Answer

The angle \( \theta \) between the lines is approximately \( \tan^{-1}(2) \) radians or \( 63.4^\circ \).

Step 1: Transform the equations into slope-intercept form

From the equation \(x + 2y = 8\), we get \(y = -\frac {1}{2}x + 4\). So, m1 = -1/2. From the equation \(x - 2y = 2\), we get \(y = \frac {1}{2}x - 1\). So, m2 = 1/2.

Step 2: Substitute the slopes into the angle formula

Substitute m1 and m2 into the formula \( \theta = \tan^{-1} ( \frac { m_{2} - m_{1}} { 1 + m_{1} m_{2}}) \), we have \( \theta = \tan^{-1} ( \frac {1/2 - (-1/2)} { 1 + (-1/2)(1/2)}) \).

Step 3: Simplify the equation

After the simplification, we get \( \theta = \tan^{-1}(2) \) in radians.

Step 4: Convert to degrees

\(\theta\) needs to be converted from radians to degrees by multiplying the result by 57.2958. So, we get \( \theta \approx 63.4^\circ \).

Key Concepts

Slope-intercept form

The slope-intercept form is a way to write the equation of a line. This form is very useful because it clearly shows the slope and the y-intercept of the line. A linear equation written in the slope-intercept form looks like this:

• \(y = mx + b\)

Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, or where the line crosses the y-axis.
To convert an equation to the slope-intercept form, you need to solve for \(y\). This means you rearrange the equation so \(y\) is by itself on one side of the equation. Let's look at the examples from the exercise:

• For \(x + 2y = 8\), we subtract \(x\) from both sides to get \(2y = -x + 8\). Then, divide every term by 2 to isolate \(y\), resulting in \(y = -\frac{1}{2}x + 4\).
• Similarly, for \(x - 2y = 2\), we subtract \(x\) from both sides to get \(-2y = -x + 2\). Now, divide every term by -2, resulting in \(y = \frac{1}{2}x - 1\).

Notice how each equation now clearly indicates the slope \(m\) and the intercept \(b\). This form makes it easier to understand the relationship between the lines, especially when calculating angles within geometry problems.

Radians to degrees conversion

Radians and degrees are two units for measuring angles. When working with angles, it's often necessary to convert between these units to better understand or solve a problem. A full circle is \(360^ ext{o}\) in degrees, which is equivalent to \(2\pi\) radians.
To convert radians to degrees, you can use a straightforward formula:

• Degrees = Radians \(\times \frac{180}{\pi}\)
• For example, if you have an angle \(\theta\) in radians and you want to convert it to degrees, you multiply \(\theta\) by approximately 57.2958 (since \(\frac{180}{\pi} \approx 57.2958\)).

In the exercise, after finding the angle in radians, which was \(\tan^{-1}(2)\), we converted it into degrees. This conversion helps make angles more intuitive, as degrees are often easier to visualize than radians for many people.

Tangent inverse function

The tangent inverse function, often denoted as \(\tan^{-1}\) or \(\text{arctan}\), is used to find the angle whose tangent is a given number. It's the inverse operation of the tangent function, which means if \(y = \tan(x)\), then \(x = \tan^{-1}(y)\).
In practical terms, if you know the ratio of the opposite side to the adjacent side of a right triangle (which is the tangent), \(\tan^{-1}\) helps you find the angle itself. This function is crucial in trigonometry, particularly when finding the angles between lines.

• For example, the exercise uses the formula for the angle between lines: \[ \theta = \tan^{-1} \left( \frac{m_2 - m_1}{1 + m_1m_2} \right) \]
• Plugging the slopes \(m_1 = -\frac{1}{2}\) and \(m_2 = \frac{1}{2}\) into the formula gives \( \theta = \tan^{-1}(2) \), which indicates the angle in radians.

This evaluation is essential when working on tasks involving the intersections and relationships between different lines, providing insights into how lines relate spatially.