Problem 43 - Algebra 2

Question

CHALLENGE For Exercises \(42-44,\) use the following information. If the graph of the line \(y=m x+b\) intersects the \(x\) -axis such that an angle of \(\theta\) is formed with the positive \(x\) -axis, then \(\tan \theta-m\) Determine the obtuse angle formed at the intersection of the graphs of \(2 x+5 y=8\) and \(6 x-y=-8 .\) State the measure of the angle to the nearest degree.

Step-by-Step Solution

Answer

The obtuse angle is approximately 103 degrees.

Step 1: Convert line equations to slope-intercept form

First, let's convert the given line equations into the slope-intercept form, which is \( y = mx + b \). This will help us identify the slopes (\(m\)) of each line.For the first line, \( 2x + 5y = 8 \):- Rearrange to solve for \( y \): \[ 5y = -2x + 8 \] \[ y = -\frac{2}{5}x + \frac{8}{5} \] Here, the slope \( m_1 = -\frac{2}{5} \).For the second line, \( 6x - y = -8 \):- Rearrange to solve for \( y \): \[ y = 6x + 8 \] Here, the slope \( m_2 = 6 \).

Step 2: Calculate the tangent of the angle between the lines

The tangent of the angle between two lines with slopes \(m_1\) and \(m_2\) is given by:\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]Substituting our slopes:\[ \tan \theta = \left| \frac{6 + \frac{2}{5}}{1 - \frac{2}{5} \times 6} \right| \]Simplify:\[ \tan \theta = \left| \frac{\frac{30}{5} + \frac{2}{5}}{1 - \frac{12}{5}} \right| \]\[ \tan \theta = \left| \frac{\frac{32}{5}}{\frac{-7}{5}} \right| \]\[ \tan \theta = \frac{32}{7} \]

Step 3: Calculate the acute angle between the lines

Now, use the inverse tangent function to find the acute angle \( \theta \):\[ \theta = \tan^{-1}\left(\frac{32}{7}\right) \]Using a calculator:\[ \theta \approx 77.47^\circ \]

Step 4: Determine the obtuse angle at intersection

Since we need the obtuse angle, subtract the acute angle from \( 180^\circ \):\[ \text{Obtuse Angle} = 180^\circ - 77.47^\circ \]Thus, \[ \text{Obtuse Angle} \approx 102.53^\circ \]

Step 5: Round the angle to the nearest degree

Finally, round \(102.53^\circ \) to the nearest whole number:\[ \text{Measure of the Obtuse Angle} \approx 103^\circ \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the simplest ways of expressing the equation of a straight line. It is given by the equation \( y = mx + b \), where:

• \( m \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, representing the point where the line crosses the y-axis.

Converting other forms of line equations to the slope-intercept form can be very useful. It allows for easy identification and comparison of slopes, facilitating analyses such as finding angles between intersecting lines. For example, by rearranging equations from standard form to slope-intercept form, like in the given exercise, we can easily read off the slope values \( m_1 \) and \( m_2 \), which are crucial for determining the angle between the lines.

Tangent of Angle

In geometry, the tangent of an angle is a fundamental trigonometric function. For angles formed between two intersecting lines, the tangent, \( \tan \theta \), is calculated using the slopes of these lines, \( m_1 \) and \( m_2 \), with:\[\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|\]This formula is derived from the properties of cotangents in triangles and helps determine the inclination of one line concerning another. When the arctangent (inverse tangent) function is applied, it transforms the tangent value back into angle form. In the exercise, finding \( \tan \theta \) was integral for calculating the acute angle between two plotted lines. Each slope affects \( \tan \theta \) differently, which aids in finding precise angles.

Acute Angle

An acute angle is an angle that measures less than \( 90^\circ \). In the context of intersecting lines, it represents the smaller angle formed where two lines cross. Calculating the acute angle involves finding the tangent of the angle between two lines and using inverse tangent functions.
The process entails:

• First, determining \( \tan \theta \) using the slope formula.
• Applying \( \tan^{-1} \) to retrieve the actual angle measure.

In the given problem, the acute angle measurement came out to approximately \( 77.47^\circ \). This value represents the sharp angle where the two lines meet, offering insight into their relative direction and closeness. Recognizing whether an angle is acute is important in diverse mathematical and real-world applications, including geometry and physics.

Obtuse Angle

An obtuse angle is one that measures more than \( 90^\circ \) but less than \( 180^\circ \). When two intersecting lines form an angle at their intersection, if one angle is acute, the other will be obtuse, as the sum of the angles around a point must equal \( 180^\circ \).
To find the obtuse angle between intersecting lines:

• Calculate the acute angle formed between the lines.
• Subtract this acute measure from \( 180^\circ \) to find the obtuse angle.

In the exercise, after calculating the acute angle as \( 77.47^\circ \), the obtuse angle was determined by subtracting from \( 180^\circ \), giving approximately \( 102.53^\circ \). This process ensures that we capture the larger angle between the lines, which can often reveal more about the spatial orientation and relationship of intersecting lines in analytic geometry.