Problem 46
Question
Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &x+3 y=2\\\ &x-2 y=-3 \end{aligned}$$
Step-by-Step Solution
Verified Answer
The angle \(\theta\) between the lines is found to be \(\arctan(-\frac{5}{6})\) in radians and \(\arctan(-\frac{5}{6}) \times \frac{180}{\pi}^{\circ}\) in degrees.
1Step 1: Find the slopes of the lines
To find the slopes of the lines, we need to write them in slope-intercept form (\(y = mx+b\)). For \(x+3 y=2\) or \(y = -\frac{1}{3}x + \frac{2}{3}\), the slope \(m_1 = -\frac{1}{3}\). For \(x-2 y=-3\) or \(y = \frac{1}{2}x + \frac{3}{2}\), the slope \(m_2 = \frac{1}{2}\).
2Step 2: Apply the formula
Once we have the slopes, we can substitute them into our formula for \(\tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2}\) to find the tangent of the angle between these two lines. Substituting gives us \(\tan \theta = \frac{-\frac{1}{3} - \frac{1}{2}}{1 + (-\frac{1}{3})(\frac{1}{2})} = -\frac{5}{6}.\)
3Step 3: Convert to radians and degrees
To convert the tangent of the angle to the angle in radians, we use the arctan function. So, \(\theta = \arctan(-\frac{5}{6})\). To convert this angle in radians to degrees, we know that \(180^{\circ} = \pi \) radians. So, \(\theta = \arctan(-\frac{5}{6}) \times \frac{180}{\pi}^{\circ}\).
Key Concepts
Slope-Intercept FormSlope of a LineTangent of an AngleRadians to Degrees Conversion
Slope-Intercept Form
Understanding the slope-intercept form of a line is crucial for various algebra and calculus problems. It is written as
\( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is where the line crosses the y-axis. To find the angle between two lines, it is important to first express each line in this form. This allows for the easy identification of their slopes, setting up the stage for further calculations.
\( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is where the line crosses the y-axis. To find the angle between two lines, it is important to first express each line in this form. This allows for the easy identification of their slopes, setting up the stage for further calculations.
Slope of a Line
The slope of a line describes its steepness and direction. It's a measure of how much the line rises or falls as it moves from left to right. A positive slope means the line is increasing, while a negative slope indicates a decreasing line.
To discover the slope, one can use the formula
\( m = \frac{rise}{run} \),
where 'rise' is the change in y-values and 'run' is the change in x-values between two points on the line. In the context of finding the angle between two lines, knowing their slopes is indispensable, as it is directly used in the formula to calculate the tangent of the angle between them.
To discover the slope, one can use the formula
\( m = \frac{rise}{run} \),
where 'rise' is the change in y-values and 'run' is the change in x-values between two points on the line. In the context of finding the angle between two lines, knowing their slopes is indispensable, as it is directly used in the formula to calculate the tangent of the angle between them.
Tangent of an Angle
In trigonometry, the tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. However, when discussing the tangent of an angle between two lines, it refers to how steeply one line is inclined from another. The formula to find this
\( \tan(\theta) = \frac{m_1 - m_2}{1 + m_1m_2} \),
where \( m_1 \) and \( m_2 \) are the slopes of the lines. By calculating the tangent of the angle, we can then use inverse trigonometric functions to find the angle itself in radians, which leads to understanding the orientation of the lines to each other.
\( \tan(\theta) = \frac{m_1 - m_2}{1 + m_1m_2} \),
where \( m_1 \) and \( m_2 \) are the slopes of the lines. By calculating the tangent of the angle, we can then use inverse trigonometric functions to find the angle itself in radians, which leads to understanding the orientation of the lines to each other.
Radians to Degrees Conversion
Angles can be measured in radians or degrees. Radians are often used in mathematics due to their natural appearance in trigonometric functions and calculus. One radian is the angle created by taking the radius of a circle and stretching it along the circle's circumference. To convert an angle from radians to degrees, use the conversion factor
\( 180^\circ = \pi \) radians. Hence, multiplying by \( \frac{180}{\pi} \) will provide the degree measure. This step is vital in many real-world applications where degrees are preferred or mandated over radians.
\( 180^\circ = \pi \) radians. Hence, multiplying by \( \frac{180}{\pi} \) will provide the degree measure. This step is vital in many real-world applications where degrees are preferred or mandated over radians.
Other exercises in this chapter
Problem 46
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