Problem 52 - Precalculus

Question

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &3 x-5 y=3\\\ &3 x+5 y=12 \end{aligned}$$

Step-by-Step Solution

Answer

The angle \(\theta\) between the lines \(3x - 5y = 3\) and \(3x + 5y = 12\) is 0 radians, or equivalently, 0 degrees.

Step 1: Calculation of the slopes

Let's express the equation of each line in the form \(y = mx + b\). For the first line, \(3x - 5y = 3\), we'll get \(y = 0.6x - 0.6\). Hence, the slope \(m_1\) of the first line is \(0.6\). For the second line \(3x + 5y = 12\), we find \(y = 0.6x - 2.4\), indicating the slope \(m_2\) of the second line is also \(0.6\).

Step 2: Application of the tangent formula

Given that \(m_1 = m_2 = 0.6\), the tangent of the angle is calculated as \(tan(\theta) = |(m_1-m_2)/(1 + m_1m_2)| = |(0.6-0.6)/(1 + 0.6*0.6)| = 0\). Therefore, \(\theta = arctan(0)\).

Step 3: Solving for \(\theta\) in radians and degrees

Observe that \(arctan(0) = 0\) in radians, which can be converted to degrees by multiplying by \(180/\pi\). This gives 0 degrees.

Key Concepts

Slope of a Line

The slope of a line is the measure of its steepness or incline, often denoted by the letter "m". To calculate the slope, we arrange the equation of the line into a common form known as the slope-intercept form: \(y = mx + b\). Here, "m" represents the slope, and "b" represents the y-intercept.
Let's review how to extract the slope from a given line equation:

• Take the equation \(3x - 5y = 3\) and rearrange it to get \(y\) on one side: \(y = 0.6x - 0.6\). So, the slope \(m_1\) is 0.6.
• Similarly, for \(3x + 5y = 12\), rearranging gives \(y = 0.6x - 2.4\), resulting in a slope \(m_2\) equal to 0.6.

The slopes determine the angle between lines. If the slopes are equal, as in this case, the lines are parallel, making the angle between them zero.

Tangent Formula

The tangent formula for the angle \(\theta\) between two lines is expressed by the equation:\[tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\]This formula calculates the tangent of the angle between lines when their slopes are known. Here are the steps to use this formula effectively:

• Find the slopes of the lines, which are \(m_1\) and \(m_2\). In this exercise, both slopes are 0.6.
• Substitute these values into the tangent formula: \( \tan(\theta) = \left| \frac{0.6 - 0.6}{1 + 0.6 \times 0.6} \right| \).
• Since the numerator becomes zero, \( \tan(\theta) = 0 \).

This result implies that the angle \(\theta\) is one where the tangent equals zero, indicating no angle between the lines—reaffirming that the lines are parallel.

Arctan Function

The arctan function, denoted as \( \arctan(x) \) or \( \tan^{-1}(x) \), is crucial for converting a tangent value back to an angle.
It is the inverse operation of the tangent function. Here's how to apply the arctan function when finding angles:

• Once we have \( \tan(\theta) = 0 \), apply the arctan function to find \( \theta \).
• Calculate \( \theta = \arctan(0) \), which simplifies to 0 radians.
• Convert radians to degrees using the conversion factor \( 180/\pi \), resulting in 0 degrees.

Understanding the arctan function can help solve many trigonometric problems involving angles, especially when relationships are depicted in terms of slopes and real-world applications of geometry. Ultimately, \( \arctan(0) \) translates a tangent value of 0 to an angle of 0 radians, indicating parallelism between lines.