Problem 47 - Precalculus

Question

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{array}{rr} x-y= & 0 \\ 3 x-2 y= & -1 \end{array}$$

Step-by-Step Solution

Answer

The angle \(\theta\) between the two given lines is approximately 0.1974 radians or 11.31 degrees.

Step 1: Rearrange equations to slope-intercept form

Rearrange both equations into slope intercept form, \(y = mx + c\) to find the slopes (m1 and m2) of both lines. The given equations are \(x - y = 0\) and \(3x - 2y = -1\). Rearranging gives \(y = x\) (slope m1 = 1) and \(y = \frac{3}{2}x + \frac{1}{2}\) (slope m2 = 3/2).

Step 2: Apply the formula for the angle

We apply the formula \(\tan \theta = \frac{m2 - m1}{1 + m1*m2} \). Substituting the values of m1 and m2 yields \(\tan \theta = \frac{3/2 - 1}{1 + 1*3/2} = \frac{1/2}{5/2} = 0.2\) after simplification.

Step 3: Convert to radians and degrees

To find theta, calculate the arc tangent of 0.2, i.e., \(\theta = \arctan(0.2)\). This gives theta in radians. To convert radians to degrees, use the formula degree = radian * \(\frac{180}{\pi}\).

Step 4: Calculate theta

Calculate theta in radians \(\theta = \arctan(0.2)\) which is apprximately 0.1974 radians. To convert this into degrees, multiply by \(\frac{180}{\pi}\) to obtain approximately 11.31 degrees.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to express a linear equation. It is written as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) is the y-intercept, the point where the line crosses the y-axis. Understanding and using this form can make it easier to identify and compare the slopes of different lines.

Let's consider how we rearrange the given equations into this form. For the line \(x - y = 0\), rearranging gives \(y = x\). Here, the slope \(m_1\) is 1, which tells us that the line rises one unit vertically for every unit it moves horizontally.

For the second line, \(3x - 2y = -1\), we rearrange to get \(y = \frac{3}{2}x + \frac{1}{2}\). This reveals that the slope \(m_2\) is \(\frac{3}{2}\), indicating that for every 2 units moved horizontally, the line rises 3 units vertically. Understanding these slopes is crucial for determining the angle between the lines using trigonometric functions.

Trigonometric Functions

Trigonometry helps us find relationships between the angles and sides of triangles, and through functions like tangent, we can determine the angle between two lines.

In our problem, after finding the slopes \(m_1\) and \(m_2\), we apply the formula \(\tan \theta = \frac{m_2 - m_1}{1 + m_1 \cdot m_2}\) to find the tangent of the angle \(\theta\) between the lines.

• The formula arises from the concept that the tangent of an angle is the ratio of the opposite to the adjacent side in a right triangle.
• In the context of lines, this formula relates their slopes to the angle between them.

Substituting \(m_1 = 1\) and \(m_2 = \frac{3}{2}\), we calculate that \(\tan \theta = 0.2\). The next step involves finding \(\theta\) itself by calculating \(\arctan(0.2)\), which gives \(\theta\) in radians.

Radian and Degree Conversion

Angles can be measured in both radians and degrees. Converting between these two is a common task in mathematics and physics.

A complete circle is \(2\pi\) radians, equivalent to 360 degrees, leading to the conversion formula:

• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).
• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).

In our example, after finding \(\theta = \arctan(0.2)\) which is approximately 0.1974 radians, we want the result in degrees for easier interpretation. So we multiply 0.1974 by \(\frac{180}{\pi}\) to get approximately 11.31 degrees.

Understanding this conversion helps in many applications, allowing seamless transition between different systems of angle measurement.