Determining the Angle Between Two Intersecting Lines at the Origin
To find the angle between two lines that intersect at the origin, we need to first determine the slopes of each line. This guide will walk you through the steps of calculating the angle between two lines defined by specific points. We will use the slopes and the formula for the angle between two lines to find the angle in degrees.
Step 1: Determine the Slopes
The slope of a line can be found using the formula:
m frac{y_2 - y_1}{x_2 - x_1}
For the first line
Points: ( (2, 3) )The slope ( m_1 ) is:
m_1 frac{3 - 0}{2 - 0} frac{3}{2}
For the second line
Points: ( (-3, 6) )The slope ( m_2 ) is:
m_2 frac{6 - 0}{-3 - 0} -2
Step 2: Use the Formula for the Angle Between Two Lines
The angle ( theta ) between two lines with slopes ( m_1 ) and ( m_2 ) is given by the formula:
tan theta left| frac{m_1 - m_2}{1 m_1 m_2} right|
Substituting the values
Substitute ( m_1 frac{3}{2} ) and ( m_2 -2 ) into the formula:
tan theta left| frac{frac{3}{2} - (-2)}{1 frac{3}{2} cdot -2} right|
Simplifying:
tan theta left| frac{frac{3}{2} 2}{1 - 3} right|
tan theta left| frac{frac{3}{2} frac{4}{2}}{1 - 3} right|
tan theta left| frac{frac{7}{2}}{-2} right|
tan theta left| frac{7}{-4} right|
tan theta frac{7}{4}
Step 3: Calculate the Angle
To find the angle ( theta ), we take the arctangent of ( frac{7}{4} ):
theta tan^{-1} left( frac{7}{4} right)
Using a calculator, we get:
theta approx 60.26^circ
Therefore, the angle between the two lines is approximately 60.26 degrees.
Conclusion
Through the steps outlined above, we have successfully determined the angle between two lines that intersect at the origin. This method can be applied to any two lines given their points.