Find the measure of angles formed by the lines 5x + 2 and 7x - 2.
Understand the Problem
The question is asking for the measure of the angles formed by the lines represented by the equations 5x + 2 and 7x - 2. This suggests a need to determine some relationship or angle measurement between these linear equations.
Answer
The angle between the two lines is approximately $3.125$ degrees.
Answer for screen readers
The angle between the two lines is approximately $3.125$ degrees.
Steps to Solve
- Identify the equations of the lines
The lines are represented by the equations given: $$ y = 5x + 2 $$ and $$ y = 7x - 2 $$.
- Determine the slopes of the lines
From the equations, we can identify the slopes:
- For the first line, the slope $m_1 = 5$.
- For the second line, the slope $m_2 = 7$.
- Use the formula for the angle between two lines
The angle $\theta$ between two lines can be determined using the formula: $$ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right| $$
Substituting our values for $m_1$ and $m_2$:
$$ \tan(\theta) = \left|\frac{5 - 7}{1 + (5)(7)}\right| = \left|\frac{-2}{36}\right| = \frac{1}{18} $$
- Calculate the angle in radians or degrees
To find the angle $\theta$, take the arctangent of both sides: $$ \theta = \tan^{-1}\left(\frac{1}{18}\right) $$
Using a calculator, we can find: $$ \theta \approx 3.125 \text{ degrees} $$
The angle between the two lines is approximately $3.125$ degrees.
More Information
This calculation shows how to find the angle between two lines represented in slope-intercept form. The arctangent function helps us convert the slope difference into an angle measurement.
Tips
- A common mistake is forgetting to take the absolute value of the difference when calculating the tangent.
- Another mistake may occur if the arctangent is not calculated correctly or if the calculator is not set to the right mode (degrees vs radians).
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