Problem 42 - 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 \(\tan \theta-m\) Find the acute angle that the graph of \(3 x+5 y=7\) makes with the positive \(x\) -axis to the nearest degree.

Step-by-Step Solution

Answer

The acute angle is approximately 149 degrees.

Step 1: Convert the Equation to Slope-Intercept Form

The given equation is \(3x + 5y = 7\). To find the slope \(m\), we need to express this equation in the slope-intercept form \(y = mx + b\). Start by isolating \(y\) on one side:\[5y = -3x + 7\]Next, divide each term by 5:\[y = -\frac{3}{5}x + \frac{7}{5}\]This gives us the slope \(m = -\frac{3}{5}\).

Step 2: Understand the Tangent Relationship

The slope \(m\) of a line is related to the angle \(\theta\) that it makes with the positive \(x\)-axis through the tangent function: \(\tan \theta = m\). From the previous step, we have \(m = -\frac{3}{5}\), so \(\tan \theta = -\frac{3}{5}\).

Step 3: Determine the Angle in Radians

To find \(\theta\), take the arctangent (inverse tangent) of \(m\). \[\theta = \arctan\left(-\frac{3}{5}\right)\]Calculate \(\theta\) using a calculator to determine the angle in radians.

Step 4: Convert Radians to Degrees

Convert the angle from radians to degrees for better understanding. Use the conversion factor: \[\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}\]Calculate this to find \(\theta\) in degrees.

Step 5: Interpret the Result as an Acute Angle

Since \(\theta\) can be negative and angles are typically positive, especially when mentioned as acute, calculate the positive equivalent. Since our problem requires an acute angle with the positive \(x\)-axis, interpret the finding in the context of a full circle, potentially using:\[360^\circ - |\theta|\]Round to the nearest degree for precision.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation provides a straightforward way to understand the properties of a line. It's expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept of the line.

• Slope \( (m) \): Indicates the steepness or direction of the line. A positive slope means the line rises, while a negative slope implies it falls.
• Y-Intercept \( (b) \): The point where the line crosses the y-axis. This helps locate the line on a graph.

When given an equation in a different form, such as \( 3x + 5y = 7 \), converting it to slope-intercept form by isolating \( y \) is key to easily identifying both \( m \) and \( b \). In this exercise, isolating \( y \) gives \( y = -\frac{3}{5}x + \frac{7}{5} \), showing that the slope \( m \) is \(-\frac{3}{5}\). This slope helps us understand the angle the line makes with the x-axis.

Angle with X-Axis

The angle a line makes with the x-axis is an essential concept in algebraic geometry. This angle helps determine the orientation of the line. For a line expressed in slope-intercept form, the angle \( \theta \) it forms with the positive x-axis is related to its slope \( m \). - When the slope \( m \) is derived from a line equation, the angle \( \theta \) is determined using the tangent function: \( \tan \theta = m \).Knowing \( \theta \) is crucial, especially for applications requiring precise directions or rotations. In the exercise provided, the slope \(-\frac{3}{5}\) gives us insight into the angle the line makes by using the inverse tangent, or arctangent, of the slope.

Arctangent Function

The arctangent function, often denoted as \( \arctan \) or \( \tan^{-1} \), is the inverse of the tangent function. It is used to find an angle when the value of the tangent of the angle is known. This function is particularly useful when computing the angle \( \theta \) a line makes with the x-axis if its slope \( m \) is given.- In our context, since \( \tan \theta = m \), we have \( \theta = \arctan(m) \). The arctangent will provide the angle in radians by default, so if you're working with slope \( m = -\frac{3}{5} \), you will use \( \theta = \arctan(-\frac{3}{5}) \).
Ensure to use a calculator that can handle inverse trigonometric functions to find this angle precisely. This step is crucial in understanding the line's orientation relative to the x-axis.

Acute Angle Calculation

After finding the angle \( \theta \), it must often be interpreted as an acute angle, especially when dealing with specific geometric problems. An acute angle is any angle less than 90 degrees. In calculations, angles derived from arctan can sometimes be negative or over 90 degrees. - To convert a radian measure to degrees, use the formula: \( \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \). - If \( \theta \) is negative or large, convert it to an acute angle by finding its positive equivalent. In some contexts, this might involve subtracting it from 360 degrees.Using this method, the angle the line makes with the positive x-axis can be accurately determined and expressed as an acute angle. This results in a clearer understanding of geometric relationships on the coordinate plane.